Hi I'm launching Syndicate Wars on Windows 7 64bit without enabling any compatibility options just launch exe after the installation and when I reach mission 2 it crashes in a way that it hangs up computer that I have to press reset button :/ Sometimes it crashes during mission 2. Any idea what is wrong?

No idea, being as the game (if you got it from GOG anyway) runs in DOSBox, so it would be DOSBox crashing rather than the game.

Will need system specs to even be able to take a guess at what is going wrong though.

I had random crashes later on using web browser or playing other games or even having computer launched with only desktop running. In system logs I found info about Tages Protection causing the system crash. I had first crash like that just after installing and playing Syndicate Wars from GOG.com but as far as I know gog.com games are drm free so tages protection installed itself along with Syndicate Wars?

You definitely didn't get TAGES from Syndicate Wars. Syndicate Wars is a DOS game (from 1996) that runs in the DOSBox emulator. TAGES requires Windows (and was created in 1999 - the first game to use it was MotoRacer 3 released in 2001), therefore it was never part of Syndicate Wars and it most certainly isn't part of DOSBox which is freeware.

TAGES files do exist in at least one GOG game (Chronicles of Riddick) but even then it is inert, being just the files alone, TAGES itself isn't installed.

Now, as you are getting crashes in other apps such as your web browser and other games it suggests that the problem is your system rather than the individual games you've played where the crash has happened. You'll need to do some testing to check that each crash is indeed the same root cause, and if it is TAGES you'll need to get that removed. I believe you can download the TAGES installer from the TAGES website and when running it use the uninstall option to get rid of it. I'd welcome advice from others on this matter, though, as I've never had TAGES on my machine.