Does multiplayer and updates work on DRM free games??

The answer is, "It depends on the game." Which game(s) were you wondering about?

Yes, drm has very little to do with that. Drm is a mehtod to control your use of the product. You will get any update from the developer (note it is up to them to provide it just like any other platform). Multiplayer is mostly online however via Galaxy or another client, not drm, but certainly online only with login and such like, and only where the game provides it. There are a few where you can play co-op offline but a very dwindling portion now as every moves always on.

In general yes. What doesn't work so well are some kind of anti cheat systems or central server architectures because you cannot identify yourself without some kind of authentication and that is basically DRM. But there are no problems, in principle.

I think the most useful for you is to list the games you are interested in and we can tell you about updates, see this thread for problematic games: https://www.gog.com/forum/general/games_that_treat_gog_customers_as_second_class_citizens_v2, and multi-player, like whether it is crossplay enabled with other shops' versions or not.

DRM and multiplayer are completely orthogonal. Multiplayer existed before DRM did. Updates predate DRM too.

Just keep in mind that some publishers / devs prefer to give us crippled versions of their games. And some tie their product to a particular service, such as Steam. Fuck them?

Many multiplayer GOG games are DRMed for their multiplayer aspect.

This may just be the first time I've ever seen this stand-alone phrase ended with a question mark. =D