Let's assume my internet connection is very slow and I bought a game from gog. My friend also bought that game from gog, but I don't entirely trust that friend.

Can I just use the files my friend downloaded?
The installer is split into multiple files. The exe-file has a valid gog-signature, but I don't know if I can check the bin-files or if the installer does that.

Yep, if you own it here on GOG, there's mp issue with you getting the files from your friend. The installer will do an integrity check, but not sure if that's enough to catch any shenanigans. Doesn't sound like a very good friend though.

Another option might be to go to a local library and DL the files through them. They'd likely have a fast connection.

Some GOG games have specific serial key for the owner [1], but offline installers are same in everyone's account.
So, who download it do not matter,

[1] https://www.gog.com/forum/general/gog_games_with_serial_numbers