Can you analyze when cloaked?
You can't start the analysis cloaked, but you can uncloak, click "analyze" and then recloak without any problems

Can you analyze while warping?
No, and if you warp while analyzing it will finish prematurely with no results

Can I dock while using probes
Yes, but you will loose all access to the probes as awell as your scan resutls and will have to launch new probes to find your signal again. So its not recommended. This goes for warping into a different system as well.

What does this "module is in use?" thing mean? Why can't I analyze?
The probe launcher must finish its launch cycle before you start analyzing. Wait for it to stop flashing green and try again

Can I analyze with multiple probes at once?
Yes, just control-click them all. Doing this in system map will also show you what your coverage is like.

Couldn't you probespam with snoops instead?
Hoshi has done the numbers and says it's no longer worth the effort. So, yes in theory, but not recommended at all.

You need a lot of probes for exploration and they're really expensive!
Well... yes, I guess. The cheapest solution is to buy the BPOs and mine your own stront, at which point they become pretty cheap. But... then you have to go ice mining. Your decision.