1 - It depends on the boat -some have very poor initial stability (they are tippy when sitting still) but great secondary stability (they are stable when moving). My Perception Sundance has great initial stability and good secondary.
2 - It depends on the boat - Your Sit in Kayak (SIK) can be fairly confining unless you find one with a very large cockpit. Most fisherman rave about the Sit On Top (SOT) boats because of the access you have to all your gear. My boat is a SIK but I like to fish early spring and late fall and the SOTs are chilly even here in KY. I trade access for being able to launch in November.
3 - It depends on the boat - (you knew that was coming) - Compared to my Jon boat my kayak is great in a light wind - it sits low and with its' smooth top sheds wind pretty well.
The one piece of advice I will give you (as will every Kayaker) is to try the boat before you spend $ on it. Almost all "real" kayak dealers will arrange for you to spend an hour or half a day in a boat. While you can get lower prices at the Mega Mart, the guy in the small shop will be a wealth of information that will be missing elsewhere. When it comes to rod holders and anchor systems and such the small shop is where the experience of the clerk will save you much more than the Mega Mart will.
Find a local shop and ask if there are any local fisherman and you will gain a great deal of information talking to locals who have "been there, done that".
Good luck,