The Imperial pilot who claims to have been on the Death Star when Alderaan was destroyed is probablyfull of shit, given that it was destroyed not long after. Then again, as a shuttle pilot, it's possible he was sent away on a transport mission soon after. note The book Lost Stars also indicated that the expedition that checked out Dantooine was launched from the Death Star, so he might have been in the crew. A couple of contingents from other ships also came for the big staff meeting Tarkin held, who later materials confirmed simply left and returned to their own ships after the meeting, most notably General Tagge, the officer who was most skeptical about the Death Star.
While it seems to be a dickish move to mock Boba, Bo-Katan and her Nite Owls understandably haven't forgotten that his clone brothers helped the Empire subjugate their planet.
If Jango Fett: Open Seasons remains intact in this continuity, the Nite Owls, having origins in Death Watch, would still hold bad blood against Boba's father, Jango, and Boba is more than happy to help finish up his late father's loose ends.
Another reason why Boba would be distrustful towards Bo-Katan is because having been living in the criminal underworld, he might have a few run ins with the Shadow Collective and later, Crimson Dawn who are ran by Maul, a previous ruler of Mandalore and temporary associate of Bo-Katan who indirectly helped Maul to rule Mandalore in the first place. So not only Bo-Katan belongs to the group his father hated but she is also the one who allowed a Sith Lord to enslave his father's people.
Bo-Katan suddenly refuses to take the Darksaber without winning it in combat, despite originally having it bequeathed to her in a ceremony the first time. Almost as if she's experienced the shortcomings of not taking it traditionally.
Given that the Purge of Mandalore happened shortly after she was given the Darksaber, Bo-Katan may feel some doubts as to whether she actually deserved to wield it. Thus, she feels the trial by combat is the only way to answer those doubts.
This has all been pretty much confirmed by The Book of Boba Fett, in which the Armorer refers to Bo-Katan as "a cautionary tale".
Luke making short work of the Dark Troopers despite the latter being a genuine threat to Din's crew (with Din barely taking out one alone) actually explains why the Trade Federation and later the Separatists would use B1 battle droids as the primary bulk of their army despite their laughable reputation as Cannon Fodder. No matter how advanced the battle droids are designed, they are still no match for a trained Jedi Master. Thus, quantity over quality was the only way for the battle droids to ever kill a Jedi just by sheer overwhelming firepower alone.
Luke's presence in this episode builds on his characterization in the Sequel Trilogy. When Luke arrives, he is shown with a practically mythical presence. He appears, saves the day, takes Grogu and is off within five to ten minutes. One of Luke's concerns in The Last Jedi is the pedestal the entire galaxy has placed him on, and the hubris of the Jedi. With this Episode, you can see where a reputation like that may come from.
R2-D2 and Grogu have a short conversation on the bridge of the cruiser, which we can't understand because R2 is beeping and whistling and Grogu is cooing, after which Grogu lifts his arms to be picked up by Luke. Remember that R2 and Grogu both lived at the Jedi Temple during the Clone Wars, when R2 served as Anakin Skywalker's astromech droid and Grogu was a youngling. So it's likely the two of them knew each other, or at least recognize each other. R2's dialogue may be him "vouching" for Luke as a worthy master for Grogu, which is what convinces him to go with them.
Din repeatedly attempts to just give Bo the Darksaber. Not only does he not want the position it grants, giving it to her is also a matter of honor: He agreed to let her have it if she helps him on this mission.
Although it may have seemed kind of lame to have Boba not be in the boarding party, instead staying outside to take out the gun emplacements, it does make sense from a writing perspective. The episode's finale would have been complicated if Luke and Boba Fett suddenly found themselves in the same room together.
Even though the Empire has smuggled so much Beskar metal from the Mandalorians, why hasn't Moff Gideon uses them as armor for his Dark Troopers? Well, the Mandalorians, the foremost experts of Beskar metal, would know ways to work around that element and Sabine has accidentally invented a weapon that can destroy Beskar.
There's also little bits and effects that suggest that only Mandalorians could make truly good beskar items; things such as Boba Fett having a ding in his helmet for who knows how long, and the beskar spear, something that culturally no Mandalorian would make, showing far less heat resistance when used against a lightsaber compared to Mandalorian-crafted armor itself.
Why isn't Axe with Bo-Katan and Koska? Because he's watching the Gozanti freighter they stole.
We re-establish that Cara's bitter over the destruction of Alderaan, and mocks the Empire's loss of both Death Stars. Why doesn't she show any signs of recognizing the guy who avenged her home and overthrew the Emperor? Even if she never set eyes on the New Republic's greatest hero in a holovid or anything, the scarcity of other Jedi and seeing him in action should have tipped her off.
Though not canon in this timeline, The Thrawn Trilogy relied on the idea that nobody knew what exactly went on in the Death Star throne room. Former Imperials believed Luke murdered the Emperor in cold blood, and thus wanted revenge. The Galaxy is a big place, and Luke was never a glory hound. While he does reach a near mythical status in both continuities, very little is known about what Luke was up to at this point, and considering he wasn't even really part of the military operation that destroyed the second Death Star, it's entirely possible that Cara didn't know or didn't care about the rumors of a Jedi among the Alliance's ranks. Especially since the Empire was able to make the Jedi seem like a myth within a generation after the fall of the Republic.
She's also probably busy picking up her jaw from HOLY CRAP JEDI! HOLY CRAP JEDI WHO JUST KILLED AN ENTIRE ARMY AND IS WALKING TOWARDS US HOLY CRAP I SHOULD HAVE JOINED THAT FORCE RELIGION!
Wouldn't you think Princess Leia, as the de facto leader of the Alderaanian diaspora, would have made sure every surviving Alderaanian knows who Luke Skywalker is?
So why can't Bo-Katan and Mando throwdown in a big spar for the darksaber? You can get a proper dramatic legend out over a nonlethal trial by combat. Bo-Katan probably can actually kick Mando's ass.
Because Bo-Katan knows his heart wouldn't be in it...and it would feel WRONG to her. She likely blames herself for Mandelore's destruction, and needs to EARN it.
Bo-Katan is likely entirely right in wanting to rip that armor off Boba's corpse. And Boba doesn't even provide evidence against the accusation on his father that he likely doesn't even know about. All he proves is that Jango digitally monogrammed the armor and by reasonable logic that means Boba inherited it. He still could have inherited stolen armor.