In the US, they're complete synonyms. You say "laid off" if you're trying to be less direct about it. It's the same as saying someone "passed away" rather than "died"
No, the US usage is pretty similar to the UK senses: no one is “laid off” for stealing from the cash register or napping on the job. They get fired.
OTOH, you could say that a division was “fired” or “laid off” and they’re almost interchangeable. “Fired” might carry the connotation that the specific division was underperforming while “laid off” could mean that it was due to some outside factor (e.g., half of R&D was laid off because manufacturing costs shot up)
Nah, we say someone got fired regardless of whether it was for cause or not. It's the go-to word for any kind of employer-initiated employment termination when you don't have to be nice about it. It's just that when someone was fired for stealing from the register, you're not going to be nice about it.
Those R&D folks who got laid off are going to go home and complain about getting fired. It doesn't imply it's their fault - it implies that they're angry about it.