yes they do, right after you spend one, they will regenerate at the speed of the server you are on dictates
like RT server will regenerate turns every 8 seconds, but you also have a Turn Capacity, which is the maxiumum storage of turns you can hold, and you cannot gain anymore than that
on RT the turn storage is 20, so if you spend 1 turn and wait 8 seconds, you will get it back, but no matter how long you wait, you wont get anymore after that
and if you are on a slower server and you may not be on to see them regenerate all the way back, yes they do come back at the same rate whether you are online or off
so yes, turns do come back when you log off