If Chewy activates Gunner at PS5 then a friendly Bandit dies at PS1, I'm assuming Chewy's pilot ability doesn't kick in, correct?

gunner is pretty explicit about what happens

Absolutely correct. If one card allows something that another card forbids the latter always trumps it.

I would imagine not. I can't think of any situations elsewhere where a rule that allows you an attack would overrule the effect of Gunner or similar additional attack/can't attack again this round stuff. It's not like you can use the new illict slot to SLAM and then expect his ability to trigger for example while you have a weapon disabled token.

I was going to copy the texts, but yeah: pretty much what the rest said.

still, chewy + gunner ain't the worst thing I can think of, very useful in the movement phase (didn't hit "untokened soonts" with the first try? well.. try again).