And ... now it's back to not working again. I can click the boxes to change the compatibility mode, but the changes won't take, and so the game crashes before it really launches.
Would it be worth trying to download C3C from Steam? Would it run on Win10?
Yes, the DL-version should work on Win10, but if you're going to go that route, I'd strongly recommend getting the DL-version from
www.GOG.com rather than Steam, since (unlike Steam) GOG still doesn't require you to DL their client software ('GOG Galaxy') in order to install/play. Galaxy
does make installing/updating your GOG-games more convenient (and has a whole lot of additional features), and is a hell of a lot less intrusive than the Steam-client (e.g. it doesn't try to phone home every time you boot up your computer) -- but if you don't want it, you don't need it.
And if so, would I still be able to play a game saved in Vanilla?
As I might have mentioned elsewhere, once or twice

although the current DL-versions of Civ3
InComplete necessarily include all the art/asset files for Civ3 Vanilla v1.29 and Civ3 Play The World v1.27 (final patch of the first expansion-pack), they only include an
.exe file for Civ3 Conquests v1.22 (final patch of the second expansion-pack), not for the earlier versions.
AFAIK, a Vanilla game-save (i.e. based on a .bic ruleset-file)
can be opened by the Conquests .exe (which normally uses .bi
q ruleset-files to generate games), but that game would then be subject to the hardcoded Conquests rule-tweaks (e.g. to the corruption rules, which make Ring City Placement pointless) and bugs (e.g. barbarian tunnel vision-bug, sub-bug) -- and if you re-saved that game via the Conquests .exe, I
believe Vanilla would then not be able to open that new save.
However,
the NoCD Vanilla .exe which you already have, can be added into the apparopriate location in the DL'd Civ3Complete installation. Having done that, you would be able to continue to play your Vanilla game(s) under the Vanilla-ruleset, if you wish to.
(Firaxis also issued a legal NoCD .exe for Civ3PtW, but you'd need to hunt around the internet for it.)