the receiving team
Yes. The ball is downed and turned over to the receiving team wherever the kicking team touches it.
Yes but only if the ball is fumbled by the other team (the runner loses possession while running or touches the football but does not catch it). The kicking team cannot recover a kicked ball if the receiving team has not touched it/fumbled. If the kicking team does touch it before the receiving team does, it is considered a dead ball and the receiving team's offense will start wherever the ball was stopped.
If the kicking team touches the ball first, they cannot gain possession. Even if the receiving team picks up the ball and fumbles it, the kicking team cannot legally recover it. The ball is dead and possessed by the receivers at the recovery point. That's right, because a punt first touched by the kicking team is always considered an illegal touch. When a kicking team downs the ball on an ordinary punt play, that's technically an illegal touch, but no penalty is assessed -- the ball simply goes over to the receiving team at that point. In the scenario presented in the question, the only difference is that the kicking team failed to down the ball on their first touch. No matter what happens after that point, they are ineligible to recover the ball.
Once the receiving team touches it becomes a live ball and if the kicking team recovers it regardless how far it went they have the ball.
No, once the ball touches the wall, ground, or runner (after 1st being touched by a defender) the ball cannot be caught for an out
Yes. If the receiving team has gained control of the ball and then fumbled, the ball can be recovered and returned by the kicking team. If the ball is 'muffed' (ball touches a member of the receiving team but no member of the receiving team has possession) the ball can be recovered by the kicking team but not advanced.
Receiving team. Same as a catch. You must have possession before its your ball.
You have 'illegal touching' on a player when a kick is touched by the kicking team. The receiving team, regardless of the outcome of the play, can choose to take the play at that spot. There is also the concept of 'touching' when a receiving player touches the ball on a kick. At that point, the kicking team can recover the kick and maintain possession. A "forced touch" is when that touch occurs because the opposing player 'forces' an opponent to touch the ball. For example, if a receiving player is being blocked by a kicking team player and the kicking team player blocks the receiving team player into the ball, the receiving team player is 'being forced to touch the ball by the block.' In this instance, the kicking team cannot recover the ball and maintain possession, because the touching of the ball by the receiving team was a 'force touch.'
As soon as anyone from the kicking team touches the ball, it is considered to be Illegal Touching. The receiving team takes possession of the ball where the Illegal Touch occurred. Just because the ball was touched, it does not mean it is a dead ball. The receiving team can try to get possession of the ball and move it forward. If the punt returner touches the football and loses possession, it can be recovered by the kicking team. However, given the way punts are covered in the NFL, this is unlikely to occur. The punt returner would likely be surrounded by players from the punting team, who typically crowd around the ball so it cannot be returned. The punt returner would most likely let the punting team down the ball and have his team take possession where the Illegal Touch took place.
Once the receiving team touches the ball it is live, like a fumble. Anybody can pick it up and run in whatever direction they like. Whoever is holding the ball at the end of the play takes possession, and yes, the punting team can score on such a play.
the defence doesnt have to touch the ball in order for the offence to get it.
As soon as the ball is touched by a member of the receiving team, the ball is live and can be recovered by either team. (If the receiving team does not touch the ball, then a player from the punting team can only down the ball, not recover it.)