Leon thinks it's Jason's doing, and he'd be in a position to have explosives prepared.
Why did everyone else who was infected turn into a mindless zombie, but Jason turned into a super-strong hulking monster who retained his mind?
It's possible that having the virus in your system but dormant causes it to 'adapt' to a host so that when it does trigger a mutation, you end up with a better, smarter monster. So someone who mutated shortly after infection (like most of the Mad Dogs) would become zombies, while someone who used the inhibitor for years (like Jason) becomes a smart Tyrant. See also Code Veronica, where Alexia spent years in cryosleep after infecting herself with T-Veronica and ended up with control over her mutant form (for a while, at least). I don't think it's ever specified what virus was used on the Mad Dogs, but it doesn't appear to be the standard T-Virus. It might even be a Veronica variant, given the spiky growths appear somewhat chitinous, like Alexia's 'armor'.
Why was stopping Jason so important? His ranting and raving about "fear" and "terror" aside, all he was going to do was reveal his monstrous self to expose Wilson and TRICELL.
Probably because he's not in the best state of mind and had a good chance of harming the politicians, journalists, and civilians.
The fact that he killed Shen May means that he's not in the best mental state. If he had gotten out, he would have killed a bunch of people before being put down, and there would've been a push for stronger military forces to stop the new bio-weapon threat, something that definitely happened in the Revelations games. It would've been totally counterproductive to Jason's goals. Of course, it would've been much better if they had ACTUALLY MENTIONED THIS.
Why did Shen May murder the sailors?
The destruction of the sub was a part of Wilson's plot to frame China (along with the hacking and the zombie attack) to kickstart a war and Jason was following Wilson's orders. Since Shen May (in retrospect) had already killed innocents (or, at least, put them in a position to be killed) while working with Jason to reveal the whole conspiracy, she probably had few qualms about killing a few more people for the perceived greater good.
Didn't Leon shoot Jason in the chest? How does he show up later uninjured?
Most likely thanks to the virus he's infected with, which he needed regular doses the inhibitor to keep from transforming.
What sense is there in covering up Wilson and TRICELL's crimes? If Leon's goal was to protect Claire, they could have put her into protective custody; he should arrange that regardless, since Wilson is alive and she's in danger. If his goal was to not take away people's hope, how does allowing Wilson and Tricell to continue experimenting on people and developing and deploying bio-weapons help anyone?
Nobody is aware of TRICELL's involvement, that's only after the events of Resident Evil 5, which takes place three years later in the canon. For the main question, I interpreted it as Leon not wanting to betray President Graham's trust. While he had nothing to do with what Wilson did his reputation and administration would be heavily damaged if it came out that his Secretary of Defense was a bioterrorist on his watch, and the President did chose to believe something unlikely purely because it came from Leon, so there's less incentive of Leon to repay that trust by leaking classified documents to someone else.
Why bring Claire into this series if she is to be given less than three minutes of screentime and have all of her abilities and capacities nerfed? She is essentially a damsel-in-distress in this series, and at most serves to develop Leon's character further rather than have any progression of her own.
