Casually I would press F5 to reload the page and start my battle again. But is this okey during a run? Since the algoritme is always the same? I don't see any rule that you can't do it but I want to be sure so my runs are still valid if I do.

Yeah you can for sure. Battles are pre-determined upon the inputs you do so if you die it's completely fine to f5 for a battle.

i would say yeah, it's what i do a lot when i try to beat difficult battles with different strategies

Reloading is allowed. The web version (and even android and offline) can occasionally crash/pause to the point where you need to reload it, and the RNG is already determined (So if you're fucked, you're actually 100% legit fucked). To add on to it, retrying battles are also allowed, sometimes you get absolutely bullshit RNG