Phew! *wipes brow*
You need to look up the specs of your laptop's graphics card, to see what is the highest resolution monitor it can support. I've no particular reason to think it would have any problems, but it sure would be a nasty shock to buy a new super-duper-high resolution screen then find out your lappy couldn't handle it.
Unfortunately (to preempt your next question) I don't know how you'd find that out. Google, I guess.
No, we can't necessarily assume that. If your laptop's graphics card is good enough, it'll support higher resolutions than the native monitor, I think.
But yeah, check.