I'm pretty sure this is going to end up under the 'it stinks, but it's not illegal' category, but here goes anyway.

End of December: Chef comes in to work WASTED, get sent home and hits my car on his way out. Hit and run. I file a police report immediately without knowing exactly who it is but with suspicions. Manager asks me not to name the guy who left, and I don't really want to stick a hit and run and DUI on the guy anyway so I comply. We settle out of court for the damages.

The guy who hit my car is subsequently promoted to manager. (!?)

End of January: I get fired for a customer getting upset because I messed up and his food was ten minutes late. I ****ed up (it happens once in a while), but the guy was being unreasonable and everyone around him knew it. Also the next night I forgot to sweep the bathroom floors. Incidentally the new manager was kind of pushing me out the door at the time because he wanted to tie one on while he closed up and it's technically his responsibility to check me out.

On my ineligibility for claims it says 'misconduct, neglect of duties and numerous customer complaints'

Can I appeal with the argument that their promotion of this irresponsible clown created a stressor in my work environment?

You can claim anything you want.

But your chances of prevailing with that reason are slim, at best.