How would you install mods on this gog version?

Should be same as for Retail version. As for GOG Galaxy version... Although it sometimes and this time too a bit differ builds, i guess Galaxy client shouldnt make differences in that. Its not force DRM, unlike Steam it not force installation be valide and uptodate/update.

I didnt try modding steam version which one GOG based, but ModManager is still there in menu. Mods are sub-folders. Place it to game root folder and if mod itself is correct (there is lua-ini files told game this is a mod and its name) - mod name appear in mod manager. Game be switched to it when applied. That work as addons. Like Winter Assault should work (and winterassault.exe still run wh40kdow.exe with WA folder prefered like TFT in Warcraft 3 original pre-Classic).

As i remember game remain in mod-state at next launch. Some main game ini/dat file store it, which folder to prefer to load. Use modmanager menu again to restore.
Did NodManager still ingame menu for offline GOG version?

The exact same as all other versions. Unlike with Company of Heroes, mods for Dawn of War 1 work for both retail and digital versions.

(for COH depend on game and game version. There is COH Online that is alternate variant of normal COH with full SP included, steam storefront editions, Legacy Edition in Steam and new Steam Edition)
(there is published mods directly to Steam - they require COH Legacy Edition).

Game itself cannot be issue. It is mods that did or not support variety of game versions (unlikely for just realised ones). By default it is like that. Most community did not switc to 1.3 versions like revievers say. Opposite - Steam version are not supported as it stated by 3rd party launcher.
So do GOG. Still shouldnt be issues. After all there is original ingame modmanager (at least in steam version there is such menu).

I didnt look to reviews first. Now did and... Gosh, someone cannot make FOG disappear. That not require mods at all. Im sure mods working. Based on how they described their issues - it is issues on their end.

I was that someone and I am developer for a major soulstorm mod so whilst I absolutely don't know everything, I do know a few things. As per our steam conversation this issue did not resolve with fog being removed in the way I was referring to.

In short, to me the Gog version is not as good for Soulstorm mods as the steam version, which incidentally is only £3 basically every other month. The reason for this is that currently there are two things that are designed to work on steam and not on gog, namely the fog removal function of mod manager and the custom race campaign addon for the unification mod. These use various injection methods that I have no idea about but the developers of those made them to work with steam.

I do own this on Gog, don't recommend it, but got it purely to troubleshoot issues for my mod and definitive answer: It works on Gog. I ended up using mod manager 2.3.2.2 with LAA set to inactive but had to manually apply the large address awareness patch to soulstorm.exe, which you can get from:
https://www.moddb.com/downloads/large-address-aware1

Make a backup of your soulstorm.exe first, then apply LAA to your original so you can restore from the copy if needed.
Fog removal does not work (technically sky removal if we want to be pedantic).