Thanks for the hidden gem of the week -- it's a little embarrassing how many times I have bought the Earth 2150 series.
Anyway, something odd is happening that I haven't seen before: on Windows 7 x64 (Ultimate RTM), GOGWRAP raises a UAC prompt when executing any game of the trilogy. Stranger still, on the first run, I got a dialog box saying Win7 automatically turned on some compatibility flags.
I hasten to state that the game works perfectly. I just haven't seen these things happen with a GOG game before. This is, however, the first GOG install I've done on Win7.
Are these issues normal?

I have Vista x64 Ultimate and whenever I've used any GOG installer it gives a first attempt to verify the file(s) and then a UAC prompt pops up. Once I OK it the file verification takes place again and then the installer appears.
So yes, I'd say it's normal.
If UAC does bother you too much, though, you can switch it off.

Thanks -- I do indeed expect UAC prompts when running an install, but this particular UAC prompt occurs when I try to run the game. That's slightly weird, I think.

Ah, I see. I've not yet tried any of the Earth 2150 games on my Vista machine yet, though I do have them from GOG. I've quite a few games in my "To Play" list before I get to them though.
That said, none of the other GOG games I've played have had UAC prompts other than during the install.
Possibly it's a Windows 7 issue. Are you logged into Windows as an admin (or even THE admin account)?
Were they installed under an admin user?
Another thing to look at is the install location. Vista protects some system folders, including Program Files, which can cause issues with older software if you install to these locations. I install my games to C:\Games now, and other applications to C:\Applications. I don't know the specifics of Windows 7 protective measures but from what I've experienced of it, it is just an improved version of Vista with extras.

For what it's worth, I get the UAC prompt as well trying to run GOG from Windows Vista SP1. It's not the end of the world and the game runs OK, but it's rather annoying.