Wanted to get one as a cheap Steam Link alternative (wayyy before that was announced) for streaming games, but that ambition was shattered when I learned what ARM structure is and how it doesn't run x86 programs.

I know it still might not be of any use to you, but there are many x86 software emulation tools available to the RasPi, including (I'm not 100% on it) DosBox. Again, I'm not sure if it's of any help to you, but I figured I'd let you know just in case. :)

More then year ago I run on Pi ScummVM and I'm pretty sure that it is from 2-3 years in repo (repository). It's cool to play old adventures on TV :)
I did not check if Pi 2 works with Scumm (no time for it) but why it wouldn't?

DosBox runs on Pi good as well!

But I don't remember how many games works on them, I didn't checked.

PiPlay (formerly PiMAME) is cool to. I tested only one game: Sonic on SNES Emulator.

When You put PiJuice (https://www.kickstarter.com/projects/1895460425/pijuice-a-portable-project-platform-for-every-rasp?ref=card) & LCD Screen on it, it could be the best mobile platform for old games ;)

Raspberry Pi is THE BEST way to make your TV media center. Everybody should check "P2P-Streams Video addon for XBMC/Kodi" with "ValdikSS Aceproxy". I was using it with old Pi and it worked smoothly, but with Pi2 it's HUGE FUN TO WATCH MOVIES/SPORTS streams!!

As for GPIO it's cool but I prefer using Arduino for mods.

It would be nice to have a small PC that I can just put in my pocket and easily bring to LAN-parties. Would it be possible to install an OS that doesn't take a lot of resources, like Windows XP or Linux on a Raspberry Pi, and play GOG-games on it?

Arch Linux, Wine and Dosbox, there you go

Those are main OS system for Pi/Pi2: http://www.raspberrypi.org/downloads/
Of course there are few more, but there is no support or upgrades are 1-2 in a year or two.

Here you are:
http://blog.petrockblock.com/retropie/

Yup, this is the back end for Emulation Station. RetroPie is great, however I couldn't get the n64 emulator, nor the MAME emulation working on it. I'll have to try again when I can get a functioning Pi2. :)

Is it possible for Windows 10 to run offline? Or is it completely dependent on the internet?

The more I read about the more I want to buy it. Maybe I'll buy it 'for my son' as his first console ?Playing split-screeen Amiga games could a a nice way of spending time :-P

The drawback of living in Poland is that Raspberries cost here double the price of US. But still it is cheap.

Oh it's a wonderful device, I highly recommend it, and it's a great way to teach him a lot of great computer skills such as coding, as well as how software works on a machine. Hmm, well I recommend checking out Pimoroni.com, you might find a kit for a considerably good price. I've had to deal with them on a few occasions and their customer service is on par with GOGs in terms of quick, helpful and friendly. :)

Why wouldn't it run offline?

I've checked how does Retropie works on Pi2 with
Dosbox:
- Freddy Pharkas - Frontier Pharmacist - works pretty well
- Leisure Suit Larry in the Land of the Lounge Lizards
- Skaut Kwatermaster
- Space Quest 4 - Roger Wilco and the Time Rippers
- The Legend of Kyrandia 1 PL

doesn't work
- Tajemnica statuetki
- The Legend of Kyrandia 2 PL

ScummVm checked and works:
- Eco Quest 1
- Kings Quest 1

Of course those games are only demos/trials etc. from internet!

In our country I preffer buing computer stuff like Pi & Arduino from aliexpress.com, maybe 1 month waiting for delivery but it's cheap (not always good! ;) ).

PS. I don't know if You know that Raspberry Pi2 doesn't like flash camera! When I started Freddy it hang becouse I flash light directly on Pi ... I forgot about it :P

As jdsgn said, the primary OS for the RaspberryPi is Linux. It won't run Windows XP at all, other than through emulation, as it doesn't have an x86-compatible processor. Windows 10 will run because it, like 8, will be/was ported to the ARM architecture (little software will run though, as almost all Windows software is written and built only for x86). Most GOG games won't run either, regardless of whether or not there's a Linux version, because they're built for x86, not ARM - HOWEVER, games running through DOSBox or SCUMMVM will run if you replace the GOG-supplied DOSBox and SCUMMVM with native ones. You might also find engine/source ports for some games - say, GemRB for Baldur's Gate and its siblings - which may very well compile and run on the Raspberry.

From what I've been reading, it looks like Win10 has some services that require to be online. I'm wondering if the OS will still properly work if there is no internet connection.