For example, if AlphaZero plays with an opponent who has a right to move chess figures any way she wants, or make more than 1 move in a turn? Will a neural network adapt to that, as it adapted to an absurd move made by Lee Sedol in 2015?
The behaviour when playing against "cheats" depends on how the agent has been trained, and how different the game becomes from the training scenarios. It will also depend on how much of the agent's behaviour is driven by training, and how much by just-in-time planning.
In general, unless game playing bots are written specifically to detect or cope with opponents that are given unfair advantages, they will continue to play in the same style as if the cheating had not occurred, and assuming that the rules are still being followed strictly. If the cheating player only makes one or two rules-breaking moves, and the resulting game state is still something feasible within the game, then the agent should continue to play well. If the agent significantly outclasses the human opponent, it may still win.
A completed, trained agent will not adapt its style to "now my opponent can cheat". An agent still being trained could do so in theory, but it would take many games with cheating allowed for it to learn tactics that cope with an opponent that had an unfair advantage.
Agents that plan by looking ahead during play can cope with more unusual/unseen game state - things that may not have been seen in training. However, they still look ahead on the assumption that game play is as desiged/trained for, they cannot adapt to new rules unless those rules are added to the planning by the bot designers. For instance if the allowed cheating was a limited number of extra moves, but only for the human player, the effects of that could be coded into the planning engine, and the bot would "adapt" with help from its designers.
[AlphaGo] adapted to an absurd move made by Lee Sedol in 2015?
Assuming you are referring to game 4, then as far as I know, AlphaGo did not "adapt" to this play, after Lee Sedol managed to put it in a losing position then it started playing badly as it could not find a winning strategy from the board positions it was in, and could not recover. I don't think any effort was put into refining AlphaGo during this game or afterwards to patch it for game 5.
"Will a neural network adapt to that ?"
The big functional difference between human mind and neural networks : human mind learns by itself, a NN not.
If we call NN the net with its layers, weights, ... this is a static system, unable to learn anything new. The back-propagation algorithm that made intelligent the NN runs outside the NN itself, in a different stage, different hardware and software, software that is not a NN but classic programming.
Thus, a NN never learns nothing while playing, driving, or any other action it is designed for.
If, in the learning stage, some cheats are done, the learning algorithm will learn and adapt to these cheats, thus the resulting NN configuration will be able to react to these cheats in the best way. But this is equivalent, in fact, to learn a different game where these cheats are valid movements.