There's also the point that Ryuga is the thirteenth Rider, and it's possible that they decided to claim that when Odin was destroyed during the series, he stayed dead. After all, Episode Final is a major departure from what we see in the show, and is depicted as an alternate timeline. It's not impossible that they would decide to make such a claim.
How were Shinji and Ren able to transform without a mirror near the end of Episode Final?
If I had to come up with an in-universe reason, I'd say that it's because the barriers between the real world and the Mirror World have shattered, so they no longer need a mirror as a conduit. But the real reason is simply... dramatic license.
Also another explanation is that they used one of the shards of the already broken mirrors; we've seen the riders transform using something as small as a motorcycle mirror, so using a broken shard from a larger mirror isn't so implausible.
Zolda officially has a Strike Vent melee-weapon card in his deck. I can understand Kitaoka not using it as he's better at long-range combat, but why didn't Gorou use it against Asakura?
I think it was because he was inexperienced. This was his first battle as a rider, and he may not have been used to using the deck. Or from a meta perspective, because it would have given away the surprise of who that particular Zolda really was.
Why did Shiro Kanzaki always apparently choose the same people to be riders. Guys like Ren, Asakura, Kitaoka, etc. Shinji, Tezuka and and the alternatives don't count because they found their own way in, but the others riders were apparently the same guys chosen by Shiro everytime. Why?
This is the good part of using a Reset Button. You get to experiment. I think at some point, he did considered other people, but felt like if it ain't broke, don't fix it.
He also gets a kind of a pass at Shinji, being someone who always somehow stumbles into becoming a rider, even when a rider has already been chosen for Ryuuki in the special.
Maybe he DID give the decks to different people in earlier time-loops, and the candidates we spend the series' run with are just the picks for his last few attempts?
Another possibility: The further back you time travel, the stronger the amnesia effect is, supported by the fact that, somehow, Yui always ends up dying in every loop. Shiro's Ripple Effect-Proof Memory only goes so far, so he doesn't realize he's somehow choosing the same people every time.
Why is the picture on the characters page from a different series? It's like using a Power Rangers picture for a Super Sentai series.
It was a picture from Ryuki. I believe it was either from 13 Riders or Episode Final.
It's just a promo pic advertising 13 Riders.
Why did Shiro have to do any of this at all?! (Spoilers for show and movie endings here.) He had the power to grant apparently any wish, create the Rider powers, everything... surely he had the power to save Yui. It's all by his own power. With a plan like "(a) create the wish, however you do that, (b) create Rider suits that can channel the power of the monsters via trading cards, (c) get 12 people willing to fight for the wish (that's a big one. Each knows there's an 11 in 12 chance of getting killed rather than winning.) (d) use Odin to win the wish himself, (e) save Yui," you would think (a) through (d) would be skippable. The movie version is much less glaring (needing Life Energy to sustain Yui and using the Rider War to pick the strongest source) but you'd think even then, as time ran down, he'd just... pick somebody.
The answer to your question is simple: the "wish" is created from the lives of twelve Kamen Riders. That is why the Rider War exists, and why Shirou needs twelve people to fight and die. Everything that Shirou creates- all of the powers and magical abilities- is a phantom that only exists in (or in connection to) the Mirror World. To create a true "wish", to create a "real" new life for Yui, he needs twelve people to die as Kamen Riders.
Also, as the movies indicate, it doesn't have to be the exact same twelve riders. In the series proper, there are 10 unique riders, but it's brought up to 13 by 2 more Odins, and a second Zolda (since the rules don't state that all 13 riders have to be active at the same time or unique). In the movie continuities, the presence of Verde (13 Riders) and Ryuga and Femme (Episode Final) indicates that 2 of the repeat Riders never became Riders (Goro never became Zolda in either movie, and Odin either stayed dead after his second death (Episode Final) or didn't die a second time (13 riders)). Presumably, his plan was to choose a few people, some of whom would get killed off early by mirror monsters (like what happened with Scissors), then the rest would kill each other off. Then he'd swoop in with Odin, who has the strongest set of powers, and have Odin beat them (or if he felt that Odin might have a chance of losing, he'd just have Odin use a time vent) - the only time there was truly a wild card was with Ryuga, who Shiro didn't plan on.
Ok, here's a good one: Seeing as Shinji is usually (That is, every time we've seen it, series, movie, special and SIC story) the biggest spanner in Shirou's plans... why doesn't he just send a monster or three after Shinji before he can even get near the deck? Or even better, send ODIN after him.
He probably doesn't expect Shinji to keep stumbling into the war. So every time he resets things back to the start of the war, Shinji just keeps ending up in the war. There's also the fact that Shiro may genuinely be unwilling to sacrifice more innocents than his absolutely has to in order to save Yui, and he knows that Shinji and Yui could become good friends if her life is saved. If Episode Final is in any way canon, he might be aware of the fact that Shinji and Yui were childhood friends and he wants them to meet again when Yui is safe, believing he might be good for her since she's fairly introverted, while Shinji is just lacking in self-confidence in some respects because he's new to the reporter game.
If the actual series is any indication, Shinji does have his limits and he was willing to participate in the War to save Yui's life. Granted Shinji got over it, but it's possible that if there was any participant in the Rider War who would've done just exactly what Shiro wanted, that is using the final wish to save Yui, then Shinji would have been the one.
In fact, in the beginning of the 13 riders special, Shinji is being attacked by a monster, there's already a Ryuki before him, and he still manages to become a rider despite all of that, it's just seems that despite his best efforts, Shirou is still unable to eliminate Shinji regardless of what happens.
I always got the feeling that that monster in 13 Riders was Shirou's attempt to get rid of Shinji.
To turn your question back on you: how can you be sure that Shirou didn't try one of those tactics during a previous time loop?
Here's another idea: Shiro does it to protect Yui. In the series he said that he gave Shinji The Survive Card since he saw great battles coming for him. Shiro did it under the guise of temptation, but in reality he predicted the actions of the Alternatives and gave Shinji the card to protect Yui. Considering that Tiger, the Alternatives and Shibaura were all after Yui in one way or another, he likely kept Shinji and Ren around to protect her.
Here's a better idea... Why doesn't Shiro turn Shinji into Odin?
He's picked homeless bums as Odin, so maybe the Odin deck needs someone easily brainwashed? On the other hand, since Shinji is not a handpicked rider, he might be impossible to mindwipe to become Odin, given his resistance to Time Vent. And then there's the danger of ending up with a variant of Ryuuga with Odin's abilities...
What did Shiro actually do when we first saw Odin use Time Vent? I kept rewatching that episode but I don't see what caused Yui to not confront him in the following loop.
To save one of their childhood drawings.
How did Asakura manage to afford having Kitaoka as his lawyer?
Kitaoka is known for always getting an innocent verdict. He probably took Asakura on as a challenge and reduced the price just for the sake of bragging rights - in fact, he let Asakura go because he got tired of the latter's attitude, not because Asakura was unable to pay him.