Hi.
I've got problem with logging in into red launcher.
I start game from gog galaxy, then red launcher oppens and when i try "sign in" button nothing happens, i've got message:
"Go to your browser to sign in..."

Depends on your default browser. It should pop up.
If not try clicking on that underlined word
It will require linking gog account to red account

Also maybe your pc has problem opening browser? Open it manually. Again, depends what default browser is.

Yup. All good
For other people , easiest way to check/setup default browser is by pressing Windows button on keyboard to pop-up start menu. And type in: browser