1. Wrong, it was already confirmed by the creators that is in fact what happened. Her memory shows him waking up after a few attempts because that is the recollection of important events... not because that is absolutely everything that happened.
2. Wrong, literally not all of the letters are past events. Tozu never said they were past events; he said they were secrets. If you want to be technical about the motive (despite the fact Tozu has already shown he is not), Desmond may have lied about not knowing the weapons were there, and it's even implied at one point by Damian. Not like it matters though, Tozu has shown to bend the truth and rules to aid the killing game. He literally said he doesn't lie then was promptly shown lying. Tozu is not a reliable narrator.
3. Wrong, courts do not require the prosecution to prove every single aspect of a case, but to prove BEYOND A REASONABLE DOUBT that only the defendant could have committed that crime. That's exactly the point made by Damian: We don't know how Eva got the weapons, but we do know that no one besides her could have done it. Even then, this isn't a court case, this is a class trial with 16 students.
4. Tools to turn the doorknob? You can turn a doorknob at home with any random thing you find. It's incredibly easy. No faulty logic by the game, just a lack of manual labor by yourself 😅