With the function "Open in LAN" you can activate cheats in Minecraft in a single player world, if you then leave the world again, but it is not next to the world "Cheats", as if you have set it from the beginning.
Is there any way to check whether the world has been opened in the LAN? Maybe somewhere in .minecraft?
The world is only available in the LAN as long as you are in this world.
As soon as you leave the world, the server is closed and the world can no longer be reached in the LAN.
This also deactivates the cheats in the world.
But can't you somehow check that? You could cheat all the items and say you farmed them yourself
What exactly do you want to check?
When the world is open on the LAN, there's a local server that appoints a player as operator (/ op). And this is the player who opened the world. Other players can't execute Operational Orders.
Gamemode can only be changed by the operator.