If I buy a new PC, will my GOG games download/install on the new PC without having to pay for them again?

Yes, as long as it is your computer. I have some games installed on both my laptop and desktop and that is alright too. As my laptops tend to break all the time, I have installed some games probably five times by now on four different machines, all mine.

Of course they will. All computer games will.

Some games in other shops come with limited downloads and you cannot save a backup copy but must redownload whenever you want to reinstall. Limits I have seen: 2 (!!), 3 & 5 times.

What shops, I've never seen anything like this...?

It has been several years and I cannot remember so well anymore since I only buy games here now but one was a dvd with activation on EA just before they made their own shop. The one with only two downloads allowed was some online shop but I cannot remember its name. Spotting that I left after having had a look at it. These numbers are from 2010-2014 all.

Unless your computer is running ARM, RISC, or some other weird CPU instructions, I don't see why not.

gyutto

From GOG Support's FAQ page:
https://support.gog.com/hc/en-us/articles/213235025-Are-my-redownloads-somehow-limited-

Gesundheit.

I have a new computer and can't see the games I bought

You must log into your correct GOG account. This one shows zero games.

If that alone doesn't work - try https://www.gog.com/user/refresh

I'm assuming, that you know that - to play your games on your new PC - you have to have them installed there first.

Where are you looking for them?

You need a more expensive computer!

Sorry, couldn't resist :P

Maybe try and login with the correct account and not the one with which you haven't bought a single game here?