I don't see any rules saying no but just to make sure.

Depends on the game, ask on the games forum/discord.

If you're using OPL for this you should always ask the mods of the game, it shouldn't always be allowed because OPL isn't really legit to begin with. However if you're not using OPL and the game has an "Install to HDD" option, then it shouldn't be a problem since it's a legit option (as in, an official "method").

I'd probably ask on a specific game's forum whether they allow such a thing. I'm not 100% certain, since I'm not familiar with them, but I'm pretty sure HDD loaders are generally banned, since they are third party software. Similarly, I think running PS3 game's with a SSD is also generally not allowed for similar reasons. Whereas PS4 SSD's are allowed since they are released by Sony. But, it's likely down to a game by game basis.