Obviously It should be banned to run a branch for the twice time and use the good of the left map. Is it the only limitation?

I don't quite understand what you mean, but if you are asking if you can prepare a branch beforehand for in Il category, no you cannot. For the run to count, it must be the first time you enter the branch.

But what about running into other branch to prepare etc?

How could running into another branch to prepare for a different one help? And time would start when you enter the other branch to prepare.

So it means the gameplay should be equal to if you start the game and direct to the playing. Entering another branch to prepare is allowed. Right?

I guess. I don't really know how you prepare anything, but I don't see why it wouldn't be allowed.

I could see crafting a sword in survival and preserving it into the arena. Other than that idk, and that likely wouldn't help a ton if at all

There is currently no known ay to preserve anything out of the lobby, so that doesn't help.