What are the rules for running on the gameboy player? Do I have to use the official software or can I run GBI for more accurate time/reduce input lag and it still be a valid run?

GBI is mostly accurate, and a lot of communities accept it. You can always double check with the mods of the game, but it’s common to allow considering how hard/expensive it can be to come by the GBP startup disc

I've been using the GBP in 60hz mode on the official PAL software and it seemed pretty much up to par with NDS, just more comfy due to being able to use the SP as a controller. Pokémon main series games seem to accept GBI and they are usually very picky about stuff like that, so it's probably fine, worst case ask a moderator or post on the forum of the game to be sure about it, if a GBA game doesn't have GBI added as a platform (like for example Pokémon Emerald which has GBA, DS, GBP and GBI https://www.speedrun.com/pkmnemerald )

Perhaps it's just an issue with NTSC consoles, GBP adds a considerable amount of input lag that GBI is supposed to remedy. But good to know some games have GBI listed as a platform. Thanks for the info!

NTSC should also be 60hz, so the same amount of input lag. I've only ever heard that GBI has better scaling, so yeah.