I don't have much interest in new games and GoG keeps recommending the same stuff to me over and over again in the newsletter. Is there a way for me to search only for games before a certain year or within a certain time frame?

Try this:
https://www.gog.com/forum/general/removed_search_page_filter_workarounds/page1

You can also browse the catalogue sorted by oldest first, that way the first bunch of pages are just old games. Here's a direct link with the games sorted that way:
https://www.gog.com/games?sort=release_asc&page=1

I can't find a "release year" filter. Oddly, I can go to a page with release year filter applied by hyperlink, but if I turn that filter off, there's no way to turn it back on. Any idea why that is? Is that a premium functionality, perhaps? :D

I also want to search for old games. My main rig is in need of repair and my laptop overheats easily. I want to buy some more old games so I can play without overheating my laptop's CPU.

GOG redesigned store search filters some time ago and now it's possible to specify Release Date range, for example:
https://www.gog.com/en/games?releaseDateRange=1980,2000

Of course collectrions containing old games will not show if you use the release date filter since the release date of the collection usually is quite recent.

Old is relative...