What happens if a player doesn't discard the extra cards (more than 7) at the end of his/her turn and non of the other players catches that player? But they do later... 1. When others have started their turn? 2. When it's the player's turn again who didn't discard, but the other player catches the extra cards then?
I'm not sure what you mean by 'catches' to me that sounds you think its the other players responsibility to stop cheating. Checking the rules from here and here (I couldn't find a copy of official game rules on a quick search) It reads that a player is not allowed more than 7 cards. This is more commonly called a 'hand limit' in other games.
There isn't anything that says other players have to 'catch them' to enforce a rule. Nothing says you can have more than 7 cards if no one spots them. This could be because either a player is cheating or more hopefully just made a mistake.
Its impossible to give a definitely answer about what happens in this game as its not covered in rules. What I would suggest is to if possible just apply the hand limit as soon as error is noticed. If this would give a player an unfair advantage then maybe discard at random although that could be unfair. The links above say the came has a quick start guide? I assume this mentions the order to do things on a turn including the hand limit? Many games come with something like this and its a good idea to use it to follow you turn. Say out loud to other players that you are discarding help each other and remind each other of rules mistakes happen less often.