As a moderator, I wonder if an Everdrive N8 cartridge is allowed to be used to play a NES or FDS game and publish the run to speedrun.com.

There is no site wide rule for them, but for pretty much all of the NES/SNES games I know allow them because the carts allow for games to be played on real hardware. In the case of NES it's the same as if you had a regular NES cart of the game.

FDS is unfortunately a bigger beast. Flash carts load significantly faster than a regular FDS so it can provide problems if you have runners on both flash carts and real FDS discs. As an example, Zelda 2 knows load times (or a good approximation) for emu, FDS disc and flash cart and they have two columns for real time and loadless time to allow for better and fairer comparisons.

I'd say allow it for NES but maybe talk to the runners for FDS and see if they have an opinion or are maybe open to helping time out some load times if the game isn't too long.

That's going to vary from system to system and game to game, as Timmi suggests.

For some systems, flash carts give an advantage you can't really account for, because of inaccuracies.

For others they're pretty much perfectly accurate.

For still more they're inaccurate but in a way that you can account for with load times.