When I try to download a game via Opera to the GoG Downloader, I get a "no registered application foe that extension" message.
Apparently the downloader should be present in Opera's programs in preferences but it's not, how do I set it up?

I had the same issue, easy enough to sort out, see attached image for where you need to be.

Go to > Settings > Preferences > Advanced Tab > Programs

In there select add

Select open with other application and browse to the downloader where you have it installed, select the file and then select ok and it will be listed in the window shown in the attachment.

Now close your browser and re open it and when you click on the file to download it will open up the GOG Downloader and away it goes once logged in.

That did it, thanks everyone. :)