Baseball statisticians—sabermetricians, as they're known—consider there to be 24 possibilities depending upon runners occupying bases and the number of outs. You can see prior data here for what is called the run-expectancy matrix of these 24 base/out states:
Some batters do, yes. That's less to do with stealing signs though, and more to do with the strategic effect the runner has on the infield (and centerfield, actually). If the bases are empty, the whole defense can focus on getting the batter out. If there's a man on second, especially if there's less than two outs, it's much more important to keep the man on second from scoring.
Depends on the batter. There's a stat for batting average with runners in scoring positions (at least one runner on at least second base).
It has a lot of factors though like the batter's own psychology, the pitcher having to pitch from the stretch (an abbreviated motion to defend against base stealing when there are >0 runners on base), etc.
Baseball stats by the actual teams to get an edge is the most advanced in all of sports. The internal stats of the teams likely have much more sohpicticated numbers that control for a lot more variables.
I'm not a huge baseball fan, but the thing about baseball that does intrigue me is the amount and depth of data captured, and all the neato things you can do with that. Someone with the right data set can absolutely answer your question, as well as come up with all sorts of other interesting correlations to predict games. It's really fascinating.
When a runner is on second the catcher will give several signs in a row and it is up to the pitcher to decode it. Sometimes the first sign indicates the "real" sign in the series, most of time time the pitcher just knows "sign N is always the real one" or something like that.
Actually the relative performance for all the different base combinations would be pretty interesting to know. Only eight possibilities!