Do I have to use GoG galaxy to play online or can I not use GoG galaxy and play online still?

If they make you use Galaxy that isn't very DRM free friendly

Online MP requires Galaxy. There's also a publisher account you can create/sign into; I don't know if it's also required for online play, though I suspect it's required if you want to play with people running non-GOG copies.

And no, it's not very DRM-free-friendly, but GOG's always been a little flexible on the "DRM-edness" of the multiplayer aspect of games they carry. The option of client- or account-based multiplayer is still preferable (IMO) to having the multiplayer stripped out, or the game simply not released here at all. (And I say this while personally having no intention of installing Galaxy for multiplayer or anything else.)

I just logged out and launched an off line game. Fun fact, I got a Steam card in the process, but I had launched the GOG version.

Silly me, I hadn't realised I had launched the Steam version. That's why I got the card drop.

I assume you mean "are made available for use in-game on other platforms, as long as you're signed in to your Nomad account", not "are permanently added (for example) to your GOG library if you have them on the Switch"? (Not that that first scenario isn't still really cool. :) )

Not sure about that. I bought the DLC's through GOG, and later I bought the base game on iPad and Android where I signed in using my Nomad account. Then I "restored" my purchases and all DLC's were added to my iPad and Android phone. The online Nomad FAQ says:

"I've purchased Talisman: Digital Edition DLC on Steam, will I have access to that content on my iOS/Android device?"

The base game must be purchased separately on each platform, but if you create/sign in to a Nomad account within the game, you can connect your purchases to that account and will be able to access them on other platforms when signed in there too. (https://nomadgames.co.uk/talisman-faqs)