I received a follow-up response from GOG Support this morning. I was asked to check that DEP (Data Execution Prevention) was on its default setting (it was), then reinstall the game and add it to my antivirus/firewall software's exception/trusted list (done; whitelisted with IoBit Malware Fighter Pro), and then install the June 2010 Direct X Runtime Package (the MS link worked; followed instructions from the suggested YouTube video). The "sea levels rising/all factions can now build Ascent to Transcendence" first-turn pop-up bug is gone from the PRACX version of Alien Crossfire, but the "hotseat multiplayer next turn" first-turn pop-up bug still appears in the other version of Alien Crossfire (the first option in the "Additional executables" tab. I assume the original Alpha Centauri game still runs normally after these steps -- but of course, Alien Crossfire is what I'd really like to be able to play again. It's unfortunate that the simple fixes GOG Support has suggested thus far haven't fixed the problem.

Ordered a new computer this weekend from Best Buy's after-Christmas sale (for reasons unrelated to this game). Will try installing Sid Meier's Alpha Centauri Planetary Pack on the new computer once I have it set up. Might try the Steam install first, since it seems to run fine (even Alien Crossfire) on my husband's computer.

Just set up the new computer, and installed both the GOG & Steam installs of the Planetary Pack. No "hotseat multiplayer" pop-up or "rising sea levels/all players can build Ascent to Transcendence" pop-up on the first turn of Alien Crossfire via either Steam or GOG -- but both still crash right after start of 1st turn. I was prompted to install DirectPlay when loading the Steam install of the Planetary Pack for the first time, and ticked the "automatically install updates" option when setting up the GOG install of the Planetary Pack. The original version of Alpha Centauri appears to run normally in both the GOG and Steam installs on the new computer. My husband (who was already able to load & play the Steam install of Alien Crossfire normally on his computer) also installed the GOG version, and was able to play Alien Crossfire on his computer via the GOG install as well.
I'm not sure what's different about my husband's computer which allows him to play Alien Crossfire normally, when my computer can't. Steam apparently would allow me to stream Alien Crossfire from my husband's computer (just a few feet away from mine), and at the moment that seems to be the only way I can think of to play Alien Crossfire.

You could probably run Alpha Centauri on a potato. You just can't run it on your current computer.

I just stumbled across this thread as I'm having the same issue -- I've been trying to run SMAX via Gog Galaxy (under "Additional Executables") and I basically can't start a new game for the same reasons: At the first turn, my colony pod disappears and thereafter I can't make any moves (and those aforementioned messages also appear).

It does appear Windows 11 24H2 is the issue, it seems to start without issue on 22H2 (granted, on a much older PC, Intel i5-2400).