I would think that it would be. The version purchased here is, after all, a legit. copy so all official patches should work.
In a worst case scenario though if there are any problems you could always just remove the mod and, at worst, the game and re-install it.

Game saves are stored in the ../Freespace/Player folder IIRC.
Backup that directory just to be sure, so you won't loose your game progress.

It isn't compatible but there really is no problem. None of the eax patches will install or change anything anyway because they don't find the appropriate executable.

The EAX patch does not apply, it can't find some file when you run it. That may indicate that the GOG version already has it, but nothing I can see allows you to enable EAX.
EDIT - Next time I need to read the whole thread, WhoKnowscs already said this.

Sorry for the thread necromancy, but as an owner of an original retail copy of FS1 I can confirm two things:
1) The GOG version of FS1 definitely does not have EAX. This can be verified by the fact the EAX patch adds itself to the version line in the main menu on a retail install. Furthermore, EAX is available as a provider on my system in the patched retail install and not the GOG version. Copying the retail executable over the GOG executable allows selection of the EAX sound provider by the GOG version's launcher. Unfortunately, it also adds back cd-checks.
2) The EAX patch will not apply to the GOG install. This is very likely because the patch doesn't copy over a complete executable but is instead a delta patch, and the GOG version is modified to remove the cd-check.
Any chance of GOG adding EAX in a future update? I'm rather fond of it.