You should read the rules from this page, sections
B. Also, I would strongly advise you to ask a referee about this just to be safe from making a mistake or from embarrassing yourself. You can come earlier before the game starts ( 1 hour will be enough for the referee to explain you everything you need to know and for you to have time to relax and psychologically prepare for the game ) and just ask him/her to explain you what you want to know ( it would be wise to write down your questions ).
I was told you can legally take your opponent's king if he fails to spot a check (and win the game in the process).
I haven't had such a case, but if the opponent plays illegal move you can stop the clock and claim a win-see the rules in the section A in the link I posted.
Also, the touch-move rule does not apply - apparently you are allowed to touch a piece, take it back and play another piece - only when you press the clock does your decision become permanent.
No, that is not the way I remember. If you touch the piece you must play with it but your opponent can hold it in the air / drag it over the board / continuously put it on the board-pick it up-put it on another square, until he presses the clock. This is very rude behavior but is allowed.
If I were to enter a rapid play tournament, what other differences from the normal tournament rules could I expect?
Read the rules from the link I posted above, and talk to the referee before the tournament just so you can be sure and safe.
This is all I know, I haven't played blitz for 11 years-that is why I have advised you to talk a little with the referee.
If you have further questions leave a comment and I will try to help.
Good luck on your tournament!
THIS EDIT CORRECTS MISTAKES POINTED OUT IN COMMENTS BELOW:
I will just quote member Remco Gerlich :
About illegal move ( emphasis is mine ):
In rapidplay, an illegal move does not lose the game. The same penalties as in slow play apply (have to take it back and move with the touched piece, opponent gets 2 minutes extra on the clock, 3rd time in the same game is game loss). The difference with slow chess is that in rapid you must claim before you make your own move, in slow chess the position will always be reinstated no matter how long ago it happened. Illegal move is only a loss in blitz chess (section B).
About making a move according to the rules:
Also, no you can't keep moving your piece until you press the clock, you can keep moving it until you let go of it.