Forget what everyone else wrote because now I am here to answer to your questions. These are the ultimate answers.
Olivera199919: I'm not quite sure how this DRM-free "thing" works. I mean , if GOG ever shuts down , would I be able to download the games I've purchased?
No, that would be physically impossible. If GOG servers were shut down, you couldn't download anything from them because the servers wouldn't be sending any IP packets to you.
However, as long as you have downloaded your purchased games BEFORE that shutdown happens, you are all good. It is the same as if you buy a CD game or a DVD movie from a store, as long as you bring it home with you and don't leave it at the store, you are all good even if that store closes down permanently.
If, however, you leave that CD or DVD into the store after purchasing it and the store shuts down, no amount of crying is going to help. You can't get that CD/DVD out of the store anymore, you should have brought it with you while the store was still open. Ok?
Olivera199919: Would I still own the games, even if I don't have a Backup? Does DRM-free indicates, that I own the copy of the product permanently, even if I don't have a launcher to download it from, or a Backup?
First you have to clarify to yourself what you mean by "owning a game".
When you purchase a game, you own a LICENSE to use that game. However, you don't own that game's intellectual property rights, as in that it would be legal for you to make copies of that game and sell that as if it was your production.
Even if the store (GOG) closed down, you still own the licenses to the games you have purchased, just like you still own a license to that DVD movie or PS3 game you bought from the Mom&Pop Video Gaming Store before they closed down. Just because the store closed down doesn't mean you aren't allowed to watch or play that DVD movie or PS3 game.
Same with GOG games. Even if the GOG store closed down, you still have the right to keep playing the games you bought from the store, as long as you have downloaded those games from the store already. The "DRM-free"-part here means that there are no technical restrictions for you to keep playing the game after the closure. On a store/service where there is DRM, you might be unable to install and play your downloaded game, a game for which you still own the license (to play).
The (closed down) GOG store, nor the publisher, is not obliged to offer you a replacement copy, in case you failed to download the game(s) from GOG before, or have misplaced the downloaded games.
Now, whether the license to a (GOG) game allows you to download a pirated copy online and keep playing that... that is a bit trickier question and grey area. A court might view it that you had a license only to the exact GOG version that you could have downloaded from the GOG store. Even if you were going to download a GOG version from some torrent site, the bittorrent p2p technology usually means you are also sharing the game to other downloaders, something that the license never permitted, so you would be breaking the license right there as well.
Nah, it is just simpler that you download the standalone offline GOG game installers before the potential shutdown, and that's that.
Olivera199919: Will the Publisher know, that I own a copy of their game?
When you buy a game from GOG, you get a receipt of the purchase to your email account. Those emails are your proof of purchase, if you ever needed to prove that you own a license to the GOG games you have on your hard drive.
I hope this is all now crystal clear, because if it isn't, this could become a loooooong night.