Can opponent move before you press the clock?

I know you are kidding, Sir, but for public knowledge's sake, THAT amounts to 'making an illegal move.'

He seems to be right, just consider the situation:
- A moves piece
- B moves piece
- A presses the clock (completing his previous move)
- A makes move
- B presses the clock (completing previous move)
- B makes move
- A presses the clock (completing previous move)
etc...

Why not? A has same right as B to move while B moved but not pressed the clock? Or I am missing something?

Well, adjusting is not a moving, so I think 6.2.d has nothing to do with moving during the other player move. That also been already discussed I think.

Here are the official rules:

https://www.fide.com/FIDE/handbook/LawsOfChess.pdf

So, I don't know what is the debate about. There is no debate here!

Article 1.1 clearly states:

A player is said to have the move, when his opponent's move has been 'made'.

Article 6.7 clearly states:

A player must always be allowed to stop his clock. His move is not considered to have been completed until he has done so, unless the move that was made ends the game.

So, if you think you can start your move on the expense of your opponent's time you are wrong and every serious arbiter will quickly penalize you for that. The rules are in place to prevent such unfair and unethical players from taking advantage. 

The quoted articles are relevant, but your conclusion is incorrect.

Note the distinction between "a move has been made" and "a move has been completed". Example:

1. The game begins, White's clock starts. White picks up the white pawn in f2 and places it on f3, but does not release it. The move has not been made yet.
2. White slides the pawn to f4 and releases it, but does not immediately press the clock. White's move has been made, so Black now has the move, even though White has not yet completed their move.
3. Black makes a move. There's nothing in the Laws that state this is illegal or penalizable.
4. According to 6.7, White can still press the clock.

People often make the IRL "premove" as the opponent still has the piece in their hand, sometimes not even on the board.