Are you supposed to remove the piece before you take it or push it off the square as you take it?

extremely hard!

Neither. You're supposed to smush the piece with your own. It adds an air of combativeness to the game.

Yes. You must make the next move with your nose.

That is crazy

Touch move rules requires that picked-up piece be captured if legally possible.

From https://www.fide.com/component/handbook/?id=208&view=article (Thanks Covfefito for the up to date link (Jan 2018) ).

Article 4: The act of moving the pieces:

4.1  Each move must be played with one hand only.

4.2.1  Only the player having the move may adjust one or more pieces on their squares, provided that he first expresses his intention (for example by saying “j’adoube” or “I adjust”).

4.2.2  Any other physical contact with a piece, except for clearly accidental contact, shall be considered to be intent.

4.3  Except as provided in Article 4.2, if the player having the move touches on the chessboard, with the intention of moving or capturing:

4.3.1  one or more of his own pieces, he must move the first piece touched that can be moved.

4.3.2  one or more of his opponent’s pieces, he must capture the first piece touched that can be captured.

4.3.3  one or more pieces of each colour, he must capture the first touched opponent’s piece with his first touched piece or, if this is illegal, move or capture the first piece touched that can be moved or captured.

If it is unclear whether the player’s own piece or his opponent’s was touched first, the player’s own piece shall be considered to have been touched before his opponent’s.

4.4  If a player having the move:

4.4.1  touches his king and a rook he must castle on that side if it is legal to do so.

4.4.2  deliberately touches a rook and then his king he is not allowed to castle on that side on that move and the situation shall be governed by Article 4.3.1

4.4.3  intending to castle, touches the king and then a rook, but castling with this rook is illegal, the player must make another legal move with his king (which may include castling with the other rook).

If the king has no legal move, the player is free to make any legal move.

4.4.4  promotes a pawn, the choice of the piece is finalised when the piece has touched the square of promotion.

4.5  If none of the pieces touched in accordance with Article 4.3 or Article 4.4 can be moved or captured, the player may make any legal move.

4.6  The act of promotion may be performed in various ways:

4.6.1  the pawn does not have to be placed on the square of arrival,

4.6.2  removing the pawn and putting the new piece on the square of promotion may occur in any order.

4.6.3  If an opponent’s piece stands on the square of promotion, it must be captured.

4.7  When, as a legal move or part of a legal move, a piece has been released on a square, it cannot be moved to another square on this move. The move is considered to have been made in the case of:

4.7.1  a capture, when the captured piece has been removed from the chessboard and the player, having placed his own piece on its new square, has released this capturing piece from his hand,

4.7.2  castling, when the player's hand has released the rook on the square previously crossed by the king. When the player has released the king from his hand, the move is not yet made, but the player no longer has the right to make any move other than castling on that side, if this is legal. If castling on this side is illegal, the player must make another legal move with his king (which may include castling with the other rook). If the king has no legal move, the player is free to make any legal move.

4.7.3  promotion, when the player's hand has released the new piece on the square of promotion and the pawn has been removed from the board.

4.8  A player forfeits his right to claim against his opponent’s violation of Articles 4.1 – 4.7 once the player touches a piece with the intention of moving or capturing it.

4.9  If a player is unable to move the pieces, an assistant, who shall be acceptable to the arbiter, may be provided by the player to perform this operation.

That would be considered an illegal move. You must use only one hand.

Ok, thanks for posting the relevant part Brendan_UK. However, the link you posted is from the 2014 rules. Not sure how much the text of Article 4 has changed but is always useful to have the latest text at hand. Here are the FIDE Laws of Chess 2018 for anyone who is interested.

Thanks Covfefito, yes always use the latest https://www.fide.com/component/handbook/?id=208&view=article (it's definitely more clear ).

Thanks, I've changed it (especially for anyone in an app that can't click on links).

Are you allowed to move a piece without physically touching it? Like using chopsticks?  Then you could claim you never touched it.

No, you can't, even though it would be funny. Chess with chopsticks! Still you could not claim you didn't touch the piece unless you say "I adjust" or "j'adoube" (or maybe if you possess some form of telekinetic powers):

4.2.2 Any other physical contact with a piece, except for clearly accidental contact, shall be considered to be intent.

How about a toothpick?

Ur supposed to knock the piece over full force with ur piece, look him in the eye, while SLOWLY placing the captured piece down on the table.

One's opponent has to be allowed to have a clear view of the board position at all times, regardless of whose move it is, so taking a piece and then deciding how to capture it is against the rules as while the piece is off the board and the capturing piece is not placed on the correct square that is not a legal position.

Something about removing the piece first seems so uppity and pretentious (in blitz at least).  In one motion I will shove the new piece onto the square while removing the old one (same hand)  This is faster for me and I enjoy the physical contact between the pieces, it is satisfying for some reason (I think I like the sound too) 

When you promote to a queen, do you "capture" your pawn with the queen, or remove the pawn first and then place the queen on the square?

I think I always push the pawn first and then replace it, the grace at which I do this is usually directly related to my clock.

they are both satisfying in different ways