Thread swapping has a non-negligible cost but I'm not sure what you mean by swapping the stack. Aren't they just swapping stack pointers? That's how I'd naively implement it.
You still have to bank a whole bunch of registers and there might be detrimental side effects but I can't imagine why the stack would get in the way. Process swapping is obviously much worse since you change the process ID and the virtual memory mapping but even then I don't see why you'd ever copy the stack.
Yes, you're right. The size of the stack is relevant to how much memory each switched-out task takes up, but not directly to how costly the switching process is.
You still have to bank a whole bunch of registers and there might be detrimental side effects but I can't imagine why the stack would get in the way. Process swapping is obviously much worse since you change the process ID and the virtual memory mapping but even then I don't see why you'd ever copy the stack.