In the game 2 of the current World Championship Series, the FIDE president Arkady Dvorkovich made the inaugural move, which differed from what Carlsen wanted the move to be. How does this exactly work, are there any rulings on this? Is it just a "ceremonial move" which is then expected to be reverted/corrected by Carlsen and if so, why does Carlsen not let him know his actual first move for him to make the proper move? From the pictures I saw, Nepomniachtchi already started thinking once he saw 1.e4, which was not the actual opening, so I don't know if this opens up for silly, minor mind games and at the same time takes the symbolic power of the move away, since it has to be corrected anyway.
How does this exactly work, are there any rulings on this?
This is not a part of normal chess. I played white in a league game last night for my club and there was no famous person to make the first move in front of the cameras. This is something special for high level competitions. So, the obvious first place to look is in the special regulations FIDE publishes for these events.
For this world championship match the FIDE document is the "Regulations for the FIDE World Championship Match 2021".
This document covers many aspects of the match ranging from how sponsors can bid for the right to host the match through to remuneration for the arbiters, the appeals committee, the FIDE press officer, etc. There is one section covering the regulation of the match itself - 4. World Championship Match Regulations.
Although there are subsections in this section covering photography, TV, broadcasting and commentary, there is nothing about a celebrity making the first move.
Article 1: The nature and objectives of the game of chess
1.1 The game of chess is played between two opponents who move their pieces on a square board called a ‘chessboard’.
1.2 The player with the light-coloured pieces (White) makes the first move, then the players move alternately, with the player with the dark-coloured pieces (Black) making the next move.
So, it becomes clear. The game doesn't begin until the first player, the player of the white pieces, makes the first move. Anybody else moving one of the pieces is not part of the game. Indeed, it seems to be part of the advertising and promotion of the event made to look like it is part of the game.
Looking further we see that if this ceremonial "first move" could be considered "advice" then the white player should be penalised.
11.3.1 During play the players are forbidden to use any notes, sources of information or advice, or analyse any game on another chessboard.
Arguably the white player taking back the ceremonial first move and making their own could be see as the more correct play.
This is purely ceremonial. Some famous person is invited to make the first move, and they usually do it without consulting the player. Why doesn't the player tell the famous person what move to make? I can only speculate, but my guess would be that it removes agency - the famous person can no longer pretend they're making a decision. It's also a small indication what opening move people like to see (although most of these ceremonial first moves are 1. d4 and 1. e4).
Sometimes you also get funny things like this:
President Gorbachev arrived at 4 p.m. and they were ready to start. Anatoly [Karpov] had white in the first game. Therefore, President Gorbachev would make the first move for Anatoly. As he arrived at the board, he told the audience that this is a wonderful match between two great champions. “I am going to cause some difficulties for the older player”, he said, then proceeded to play 1.g4 without asking Anatoly. Both Anatoly and Susan [Polgar] looked up at President Gorbachev in shock. Anatoly thought that the President was joking as Anatoly mentioned that he would like to make another first move. But the President was firm in saying there is no take back in chess. Then the players started their first game.
(For context 1. g4 is one of the worst possible first moves for White)
Needless to say, this won't happen in serious games.