Can't they just pick it up and start running? I get how the punting team want to get as close to the end zone as possible before touching the ball to make the play dead but I don't get why the returning team can't impede them trying to do that.

Well they can but the problem is the second a returning team player lays a finger on the ball (or it touches any part of their body) it becomes a live ball. Meaning the kicking team can pick it up to regain possession. During a punt there are several players from the kicking team flying down the field towards the ball carrier. If you let it bounce and then pick it up by then they're likely right on top of you and the ball.

There's a few other nuances I haven't quite got my head around. 1.) If a receiver drops a thrown catch it's an incomplete pass. Is that the same on a 4th down? On a 4th down, would a drop result in the defensive team getting the ball at the line of scrimmage?

Yes it's a turnover on downs.

2.) If the offensive team had've decided to punt on the 4th down instead of throwing it, and the punting team still drops the ball, is the ball live and can the returning team pick it up and run? Or is it considered dead because the punting team touched it?

Yup like I said earlier it's anyone's at that point. Dropped punts are usually referred to as muffed punts, though like I said it doesn't have to be dropped - if the edge of your foot so much as touches it it's a live ball. Sometimes you'll see players on the return team trying to avoid the ball so they don't accidentally touch it while the kicking team is near.

4.) If the returning team touching it makes the play live, why don't they just jump on the ball to stop it rolling further to the end zone? Or can they only touch it while it's in the air, otherwise it's some kind of penalty?

I suppose they could do this, but as you'll learn footballs seem to have a mind of their own when they're on the ground. Players will jump on the ball during fumbles and the ball still seems to squeak out sometimes.

One thing we hear a lot about down here in Australia is punting. We've had a few ex-aussie rules players go over and try their hand at punting. They always talk about hang time and how important it is. Is it important because you want the punting team to put as much pressure on the returning team so they might drop the catch or decide not to catch it at all? And... to also prevent the ball rolling over the end zone and hopefully stopping it just short?

Well ideally hang time would be so you (The kicking team) could reach the spot of the ball and stop a return or try to down the ball before the end zone. As you said earlier if the kick goes in the end zone it's a touchback. So a lot of times they'll try to "down the ball" inside the five or ten yard line, keeping it out of the endzone so the offense will be backed up.