I was playing in a tournament once, during a sequence of captures, my opponent could recapture my Knight about 4 different ways. He just removed it and carried on thinking.
What should I have done?
The mere act of touching one of your pieces obligates your opponent to capture it (if legally permitted) on his current move (at least according to USCF standards), unless he explicitly declares his intent to adjust the piece beforehand. Assuming the clock continued running on your opponent's time and he did eventually choose how to capture the knight, I suspect that what he did would be considered legal (though somewhat unorthodox). However, if he didn't end up properly capturing the piece after touching it and before punching his clock, you would've unquestionably been justified in complaining to the official arbiter.
If you are playing with clocks and he sticks to the rules then it is fine. So if he takes the knight off the board and can capture it then the rules oblige him to do it. The move is finished when the capturing piece is placed on the square and the clock is pressed.
It is a bad habit though. Normally, you shouldn't touch any piece on the board until you have decided which move you want to make. If he removes the knight first, he is obliged to take it. Thinking about it afterwards could lead to the conclusion that taking the knight is not the best move after all, but it is already too late.
Another bad habit is the "eagle claw". The players is about to move, but continues thinking while his hand hovers over the board, sometimes for minutes.
Best thing is, think about your move, make a decision, write it on your score sheet and make the move without interruption.
According to the rules it is allowed to first remove the opponent's piece and then move your own piece to that square and then press the clock. Ideally, placing your own piece should be done directly after removing the opponent's piece. It is a bit strange to first remove the opponent piece and then sink down in deep thought. This is not considered as good manners. Yet the rules do not seem to prohibit this method of capturing.
If this happens to you again, you can consider calling the arbiter and say that this type of capturing is disturbing. It is not allowed to do things that disturb your opponent during play (talk, each chips and spill some of them on the board, make funny faces, click a million times with your pen, etc.). The other option is to ignore this and focus on your own game.
There is no time limit (except for the over all limit) on any one move. If you have two hours to make X moves, you can take one hour and fifty nine minutes to make one move, and use the last minute to make X-1 moves.
If you touch one of your pieces, you must move it. If you touch one of your opponent's pieces, you must capture it. (In both cases, it must be legal for you to do so). If you have multiple possible moves or captures, you can take as much time as you want to think about it, subject to the time constraints above.
But it is "unprofessional" to start a move and take a long time to finish it.
Depends upon how big a fuss you want to make over it. I’m less sure than some others here that it’s legal; there’s enough grey area around it to have grounds for a complaint, though not enough to be sure you’ll win the ruling.
It flows from the fact the rules specifically state captures are made by moving a piece unto a square occupied by another, and then immediately removing the captured piece from play (7C USCF, Section 4, I think, FIDE).
Your opponent clearly is not simply touching the piece, he is removing it from the board. As such, he is performing the actions out of order. That can be the first part of your complaint.
Under USCF rule 20G, players are prohibited from conduct that annoys their opponent. TDs/Arbiters have discretion as to what can be considered annoying, but pairing the out-of-order factor above with the fact you find it quite disconcerting to be staring at an empty hole where your piece was could carry it through.
For added weight, you might add the claim that your opponent’s early removal of your piece constitutes an illegal aid to his analysis, since he is, in effect, using a physical view of the chessboard that does not reflect the current position while thinking about his move.
I’m not going to claim those three points taken together will guarantee the Director/Arbiter will side with you, some will and some won’t, but if you find the behavior annoying enough, it might be worth a try next time.