Punt Near Endzone

[MD Longhorn]

You don't mention what ruleset, so mileage may vary, and I'll specifically state that NFL rules are completely different from what I'm about to say.

NCAA rules (and I believe Fed is same or close):
All that matters is the ball. Period. Nothing else. A player laying completely in the endzone that stops the ball from crossing the plane of the endzone is not a touchback. You ask about possession - if the ball is possessed by the kicking team, the play is dead. If not, it's not. So you could have multiple touches by the kicking team - the receiving team (assuming they don't eventually pick up the ball and then get a penalty) will get to pick the most advantages spot of the many touches. Also, if the ball remains motionless and no one is attempting to recover it, it's dead at that point as well.

Hope that answered your question.