Weird player movement
In general, weird player movement is covered by Article 4: The act of moving the pieces. Note that if an arbiter observes any volation to Article 4, they must intervene immediately.
Handling illegal moves and illegal move claiming
Illegal move claims are dependent on when the move has been made or completed:
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If a player picks up a piece and makes a move and releases the piece (and doesn't press the clock), if that move is legal, then that move stands. The player cannot change this move. This is covered by Article 4.7.
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If a player picks up a piece and makes a move and releases the piece (and doesn't press the clock), if that move is illegal, then the player can still change the move to any legal move. Touch move still applies (unless of course, the illegally moved piece cannot be moved anywhere legally, then any legal move can be played). This is covered by Article 4.7.
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If a player makes an illegal move, the opponent cannot claim it until the player completes the move
Questions
Q: A player picks up a king and moves it two spaces to castle. The player presses the clock without touching the rook. What do you do?
A: Illegal move. Illegal move penalty applies. The king MUST do that castle, they CANNOT do (e.g. a long castle the other way). They CANNOT do a normal king move. For multi-part moves (e.g. a move with two actions, such as castling, capturing, promoting), you MUST do that multi-part move even if you've only done part of it.
As per Article 4.7.2, and Article 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.