games downloaded by gog galaxy
does the game require galaxy to be running while i play or can i turn it off after it launches

No Galaxy needed.

depends on the game. Some games (like Cyberpunk or No Man's Sky) lose some content without galaxy connection.

I think NMS was fixed. Single player games don't need Galaxy to run, some MP modes do.

Beware that if you install a game using Galaxy, the shortcut will be set to "Launch Galaxy and tell it to launch the game". So you should make your own shortcuts (or run from the exe) if you don't want Galaxy running.

A costume is not going to make a difference in cyberpunk.

Quite true. Some games do require galaxy, like cyberpunk for some small content, gwent completely needs it, absolver is mostly online so requires it. Witcher adventure is also mostly online so needs it. Almost all multiplayer needs it. Also, even if you download the offline installers these still contain components of galaxy, and the installer is now setup in galaxy style, so not a simple unpack.
In terms of what you downloaded, did you download the offline installers? It will be an exe plus possibly some bin files if so. Those you can save on backups or copy elsewhere. No, the game is so broken and poorly designed from the ground up, several gigs of patches is the only thing that could save it. Doesn’t of course excuse the online gated content, regardless of how much it is, one costume, one level it’s irrelevant, it’s still online gated content, and someone might really want it.

That was intentional, of course. GOG is convinced most people like gog galaxy, which is why they keep leaning further from DRM-free.