GOG and Steam pages clearly say it works on XP.

Classic installer game doesn't work for me though all my specs are more than sufficient. Game just doesn't start at all.

What is this "g" in 1.6.2g version indication? Could it be possible that GOG placed Galaxy version as Classic by mistake (or - God save us from it - by some mischievous intent)?

I may be about to but this game and dlc's, so I hope the error is yours not GOG's. Yes the "g" does represent Galaxy files, so when you install the game Galaxy will install as well (unless it already there). Are you sure you both:
1) Downloaded the offline installer(s)
2) when starting the install, go to options and make sure galaxy ins't about to send you to the dark side.

With the "g" suffix you do have the Galaxy files and Galaxy doesn't support XP.