Am I allowed to modify my weapon? I didn't see it in the rules.

Sorry, the notification for this must've gotten lost among submission notifications!

Nothing that requires outside modification of the game is allowed in any category.

A New Game run must start from a clean save file, nothing unlocked/edited. Basically you should think of it as "delete your save file before running this".

A New Game+ run you can use whatever you have unlocked/upgraded.

Looking back on this, I should've clarified: in an NG run, you're allowed to upgrade your weapon during the run with the points that you get.