After sliding the player "may pass, shoot, start a dribble or call a time-out. Once he/she is no longer sliding, he/she may not roll over. If flat on his/her back, player may sit up without violating. Any attempt to get to the feet is traveling unless he/she is dribbling. It is also traveling if player puts ball on floor, then rises and is first to touch ball." (taken straight from NFHS Casebook 4.44.5 Situation B)
In your last question, if player is sliding and no TO called, but player has possession, there has been no violation. I suppose, in theory, a player can just lay there with the ball after sliding for an undetermined amount of time, unless subject to some other requirement, such as 10-seconds if in back court, 3-seconds if in lane, or 5-seconds closely guarded. Most likely, the next whistle will be for a held ball, when the opponent decides to grab it.