Now, Ezio appearing in Soul Calibur V may seem right, yet one must note that Soul Edge (the first game) is set decades after Ezio's death (Siegfried was born almost a hundred years after Ezio was born). Brilliance kicks in when you realize the game is actually a program installed in the Animus, hence explaining the "glitches" experienced during Ezio's debut trailer. Besides, what better way to train Desmond by having his ancestor fight samurais, ninjas, demonically-possessed swords and gods? Take note that this kinda allows his appearance in the game to be explained that it is non-canon, as Ezio is already dead by the time SCV rolls around.
Oddly enough, according to one interview, "the power of the Animus" was actually Ubisoft's suggestion, while a Soul Calibur community manager in a separate interview noted that there were several undocumented gaps in Ezio's history during the events of Brotherhood (the incarnation used in SCV), suggesting the possibility of Ezio himself traveling in time/space to the Soul Calibur universe.
I didn't realize Soulcalibur was a portmanteau of "soul" and "excalibur" until just a couple of months ago. The sudden realization occurred while I was walking home from school, and the face I made was probably the most accurate representation of this thing◊ ever. -Crazed Ninja
"Soul Calibur", literally, means "spirit blade". "Soul Edge", literally, means "spirit blade". Well, that certainly explains the home title of the original Soul Edge as "Soul Blade", but why do these two swords, clearly completely at odds with each other, have the same name? ... Is a question that was perfectly viable when Soulcalibur II, or III, were the most recent. Then IV comes along and proves some people of being right with their Fridge Brilliance - Soul Calibur has perfectly good reason to have the same name as its nemesis. Not only is it made from a shard of Soul Edge, thereby technically BEING the same sword, but as Siegfried's ending shows, Light Is Not Good - Soul Calibur has the same desires as Soul Edge, it's just made specifically to destroy Soul Edge, so it does so, while at the same time fulfilling its own goals. It's better than Soul Edge by virtue of it being the smarter one, but not by much.
In a way, Soul Calibur is the exact opposite of Soul Edge, right? In the earlier games, that seemed to imply since Soul Edge was Always Chaotic Evil Soul Calibur was automatically good. However, Soul Calibur doesn't represent good, it represents Order. And you have to admit, a world encased entirely in crystal is pretty orderly...
Once Soul Calibur's origin in the fourth game is revealed, a lot of characters past actions start to make sense. Algol created Soul Calibur using a piece of Soul Edge, with one purpose: to destroy Soul Edge. It does this the only way it knows how: by picking a host to wield it, and manipulating it into doing what it wants.
Pyrrha's theme is entitled "Wings of Sorrow." This mirrors Tira's stage theme in her debut appearance, "Wings of Despair." Pyrrha's Japanese profile mentions how she continues to fight due to fear that Tira will leave her and she'll be alone again. This mirrors Tira's reason for serving Nightmare, fear of being abandoned, as seen in her SCIV ending.
Thematically, it also ties in with one of her mother's themes: "Wings of Faith" from the original Soulcalibur.
Alpha Patrokolos and Pyrrha Omega's names have the Greek characters appear different order. This is because Alpha is the first letter of the Greek alphabet and Omega is the last. And maybe because Patrokolos and Pyrrha represent the beginning and the end of the Soul series' main conflict because of how they were hosts for Soul Calibur and Soul Edge but ended up destroying both swords.
Christian symbolism aside, that would've made more sense if the conflict between the soul swords didn't begin all those years ago with Algol and his son. The end is debatable as well, seeing as Daishi wants to churn out sequels at least up to SCVII. Even when the swords seem to be destroyed or sealed away, they always somehow come back for more. I believe the Word of God for that naming convention (if I'm not getting my sources wrong) was that Alpha Patroklos uses the original style he learned before trying to imitate his mother's moves whereas Pyrrha Omega represents the finalization of her fighting techniques, thus corresponding to the Alpha and Omega in their names (which, as you said, are the first and last letter of the Greek alphabet).
Another possible interpretation/meaning: Sophitia wasn't the first warrior Hephaestus recruited. He recruited 24 warriors in all, corresponding to the Greek alphabet. Sophitia was the last one; "Omega." Now both Patroklos and Pyrrha are carrying on the legacy of not just their mother (Pyrrha Omega), but all of the Holy Warriors (Alpha Patroklos). Though this interpretation is kind of tampered by the Japanese sourcebook confirming that Hephaestus has created a new generation of the 24 Holy Warriors himself...
Another answer could be simply this: Soul Calibur is seen as a helpful weapon whose existence offers hope for rebuilding and banishing the threat, the beginning, where Soul Edge is supposed to be the destroyer of all. Thus, Calibur would be seen as Alpha the beginning, a way to keep going, the start of a new journey (i.e. prevention of destruction caused by Soul Edge), and Edge would be Omega, the end, a means to stop all journeys, a way to stop those you dislike.
Yoshimitsu the Second has two taunts in SCV. The first is, "It is I, Yoshimitsu!" The second is, "Evildoer... you shall repent!" The brilliance here is not only does he use the correct taunts for obviously evil characters, such as Nightmare, Tira, etc, but also for Patrokolos. Why is this brilliance? Because in the story mode, Patrokolos was slaying innocents left and right, believing them to be malfested. Yoshimitu's mission statement takes offense to that, to say the least.
It's also a call back to Soulcalibur II; Yoshimitsu the first said both the above (or close variants thereof) as his pre-fight intros.
As of Soulcalibur V, the Korean characters have all apparently left. Of course, now that they know both Soul weapons are evil it makes sense how they aren't interested in pursuing either of the swords anymore.
Wait, what? Why are Koreans the only ones with common sense now?
Word of God states that since the Japanese invasion ended, the Koreans (Yun-seong in particular, as Hwang and Mi-na knew that the blade was evil) had no further need to seek out Soul Edge. The three were heralded as heroes upon their return to their homeland. However, they didn't know that Soul Calibur was just as destructive as Soul Edge; so far the only characters who came to that conclusion were Taki and Talim (the former barring witness to Siegfried and Nightmare's battle at Lost Cathedral in SCIII and barely escaping the aftermath with her life, the latter seeing it in a vision).
Cassandra figured it out too, and does not return for V. This one's pretty solid.
Not exactly. She only figures out that Soul Calibur is evil in her ending from IV. Since she was canonically knocked out by Sophitia at Osthreinsburg, only to awaken and then be flung into Astral Chaos, Cassandra likely never came to that revelation (and even if she did, she's currently trapped somewhere in the other dimension unless she can find an astral gate before Kilik closes all of them for good).
Edge Master randomly hanging out in Astral Chaos seems to be very odd. Until the artbook revealed than Night Terror is loose in that dimension, and could likely, at any point attempt escape (it doesn't actually think, which may be the only reason it does not), as someone far weaker (Pyrrha) cut the dimension open. Edge Master is in there to make sure Night Terror stays there. Crosses over into Fridge Horror when you also realize Cassandra is trapped in a world with an insane berserker out to destroy every soul it finds. Even Algol seems to keep his distance.
Cervantes, Kilik, and Siegfried have also passed through Astral Chaos at varying points in the series (although Siegfried doesn't count seeing as he was Nightmare at the time and therefore Night Terror wouldn't have existed at that point) without encountering it. That also raises the question of what Night Terror exactly is, seeing how it was Nightmare's One-Winged Angel in III but V's artbook indicates it's something completely different, despite it wielding Soul Edge (Complete).
Edge Master is officially there to deal with Algol (The Man Behind the Man in V), who is trying to Take Over the World by corrupting the world with the energy of the soul swords. He's probably aware of Night Terror's existence, though, seeing as he officially sensed Angol Fear coming to Earth according to the relationship chart in IV (or rather, Broken Destiny).
This may be a phenomenal Hand Wave, but Soul Calibur's increasing and largely out of left field Knight Templar qualities in IV and V might make more sense when you realize that the sword is (for the most part) no longer accompanied by the other two treasures. The Dvapara-Yuga is meant to purify Soul Calibur, while the Kali-Yuga channels and cleanses any negative energies that might have incurred within her over the years. Soul Calibur, when she was disguised as the Krita-Yuga, was in close proximity to the Kali-Yuga and Dvapara-Yuga and routinely underwent purification rituals. The same presumably holds true of when Kilik and Xianghua traveled together in Soulcalibur. Without the constant checks and balances imposed by the other two treasures, Soul Calibur, instead of being an unambiguous force for good, reverts back to being the lesser of two evils between herself and her brother (thanks to the unwitting taint of Algol's soul when he sacrificed his life to forge Soul Calibur), and not by much. This is why Talim had a vision before IV of Soul Calibur becoming more and more malevolent; it took a while for Soul Calibur/Elysium to regress to her original nature. Being trapped in the Soul Embrace with Soul Edge for all of III might have also served to expedite this: Soul Calibur was already corrupted once by Soul Edge due to being stuck with Inferno in the void for all of those years (remember Soul Calibur (Evil) from SCII?).
Kratos battles Sophitia in Broken Destiny's opening movie, and even has a unique quote upon defeating her with a Critical Finish. Brilliance kicks in when you realize that Sophitia is an Athenian, and given the "differences" between Spartans and Athenians as well as Kratos' own relationship with Athena...
Zasalamel is, at least in Fanon, assumed to be Mesopotamian, and his weapon is a crescent-shaped scythe. Mesopotamia is also known as the Fertile Crescent.
Link just appears through a time vortex and leaves suddenly after having completed his mission in II. Take a look at his main game counterparts from the eyes of a Hyrulian historian, or even better, the legend of the Hero of Time at the beginning of Wind Waker, which was released before this game was ported to the consoles. Most incarnations of Link just seemingly appear from nowhere and disappear from the word or fade into obscurity after the events of their game's end, exactly like this Link.
There's a lot surrounding the Star Wars characters and their weapons... so we'll start there.
It was lampshaded in a promo comic, where Starkiller fought Taki and was shocked his lightsaber didn't instantly destroy her weapons.
Pyrrha wasn't returned to her family and grew up without knowing of them... yet she somehow uses her mom's fighting style (with a bit of Cassandra's too since both of Sophitia's kids in V will be using bits of both Alexandra sisters styles) how could she know this style without growing up with them. her brother Patroklos probably can be hand-waved off as being raised by Cassandra after Sophitia dies/vanishes (not clear what happened officially yet) but not her.
It might be implied that there's something about the weapons themselves that gives her the knowledge to fight. Her profile describes her discipline as "protected by her sword and shield."
Sophitia AND Cassandra's sword skills were both given to them by their sword... which Pyrrha now uses. Patroklos actually used another style (Setsuka's) but started to imitate what he thought was his mother's style (his Sword and Shield style is described as self-taught). Thus he is not only the only member of the family that actually learned to fight on his own... he also did so on his own.
Possibly Fridge Brilliance: Sophitia was ridden with shards of Soul Edge when she broke the first sword in the battle against Cervantes. Taki was able to surgically remove all except one, as she would risk killing Sophitia in the process. Pyrrha's malfestation is the result of absorbing the energy from this fragment while in the womb. Despite losing Soul Edge after the very first game, Cervantes has never lost access to his supernatural moveset, the reason being that these techniques are the memories of Soul Edge. It could be explained that Pyrrha subconsciously recalls some of her mother's fighting style in this same manner, though she remains a reserved fighter - this doubly reinforces the theory, as her fighting style is extremely close to Sophitia's when fully awakened as a Malfested, thus tapping fully into the latent memories of Soul Edge that she absorbed as a fetus.
Plot Hole with SCV's story. In Cassandra and Sophitia's character profiles for SCII and SCIII, it's mentioned that BOTH of Sophitia's kids were tainted by Soul Edge. But by the time SCV comes along, Patroklos has no physical qualms with wielding Soul Calibur despite being born with a taint he should be physically adverse to or that said taint shouldn't been enhanced by his not-so pure intentions and actions. Did they retcon something and only include it in the artbook?
There has been no retcon to Patroklos' malfested status itself as Elysium's profile makes mention of the "evil energy" that is inside his body. According to her profile, while he has a "small amount of evil energy" in him, his "sense of duty" still made him a viable host in Elysium's eyes. The implication appears to be that since Pyrrha was born first, she inherited more of Soul Edge's taint than her brother did, which made her a more viable choice for Soul Edge.
Is this stated in her Online Profile or in the Artbook? SCV itself doesn't have any profiles to go off of. Also, how did Patroklos "small amount of evil energy" not increase from his less than heroic actions when this evil energy is explicitly shown and stated to feed off the host's inner sinful deeds? "Sense of Duty" doesn't exactly seem like a good counter for explicit gleeful acts of vengeful cold-blooded murder.