I have not used it myself but you should be able to run typical linux desktop software with steamos. That being said, I don’t think there’s gog galaxy for linux. You can use HeroicLauncher for GOG (and epic and prime gaming) instead.
I think this post provides a pretty good high level introduction: https://www.gamingonlinux.com/guides/view/how-to-install-extra-software-apps-and-games-on-steamos-and-steam-deck/
Alternatively, you could try a gaming focused Linux distribution like bazzite (based on fedora/ublue) which comes with a lot of gaming related stuff preinstalled or easy to install. The base system is mostly read only though and if you need the latest version of some software that’s not on flathub, it can be a little limiting.
cachyos (based on arch) is a good alternative if you would like to be able to get your hands dirty tweaking your system or want more freedom wrt installing software.
In any case, you should check https://www.protondb.com/ if your games are supported.
Some games with certain anti cheat software can not be run (e.g. anything using vanguard, riot’s anti cheat).
Do note that if you’re in the EU (or can convince Windows/Microsoft that you are, I assume) you should get Windows 10 patches for another year without doing anything.
What I’m reading is that you want site-to-site connectivity. Wireguard + possibly dynamic DNS makes this pretty easy (assuming you can open ports and configure NAT at your sites). Or you could set up some other VPN solution like OpenVPN.
There’s also tailscale (a paid service) for facilitating the wireguard setup, NAT traversal and relaying. headscale is a self hosted solution that aims to provide something similar (but more limited in scope).