What happened to Tom Robinson, and why did he run when he still had the chance of an appeal?
Tom was shot 17 times after he allegedly tried to escape from the prison. That many shots would certainly kill a person if any of them hit the right organs, so yes, he died.
We don't know if he did run, that is just what was reported by the guards. If he did run, and if he was indeed trying to escape, he likely thought death a better option than being stuck in prison and had lost hope in the appeals system after he saw his own obvious trial play out in from of his own eyes.
If he didn't run, I think we can chalk his death up to a race issue. This is one of those missing scenes in a book where we hear a few details and have to infer or speculate about all that could have happened based on what we know.
My gut tells me Tom didn't run thinking he could escape for three reasons: 1. he is moral, 2. he trusts Atticus' ability, and 3. he is crippled.
What happens to Tom Robinson is that he is killed while trying to escape from jail. He just runs towards the fence in full sight of the guards. It is as if he is trying to get killed.
Tom does this because he figures he is probably going to get killed anyway or at least sent away to prison for a very long time. He feels that the whites control the criminal justice system and a black man like him will not be given a fair appeal hearing any more than he was given a fair trial in the first place.