When I want to sign a FA during the season, is there a way to cut a player only if the wanted FA is signed to my team? Or do I have to cut a player to make room, and in the case where the FA signs with another team, I am left with an open roster spot?

You can sign the player without dropping another if you're going into a midweek stage. You then drop a player going into game day. If you're going into game week while trying to sign a player, you must drop a player as well.

... or the AI will drop one for you, probably your most recent first round pick!

(This actually happened to me. The AI cut my 1st, 2nd and 3rd round picks because I was over the limit for training camp. Of course they were quickly scooped up by other owners.)

Probably the worst thing that can happen to you is the AI having to cut players to get you under the cap. He tries to do it by cutting the fewest players possible, so the cuts are in order of the highest base salary until you're under the cap with enough room to get you back up to the 46-player minimum with min-sal players.

Thanks!