What worked ok for me: instead of setting the "Windows XP SP3" and the "Run As Administrator" flags on the shortcut, I set them on the executable directly. Now I don't need to remember to start the game or CivAssistII as Administrator, Windows always asks me automatically.
However, one thing is strange: everything shows fine in CivAssistII, the trade options, military, culture values, flip risks, etc, but it doesn't display the map. Looks like it cannot find the graphics files for terrain?!
And once in a while there is a .sav file, which it doesn't understand. If I copy the exact same .sav file to my Win7 computer, it loads fine?! Ah, and the auto-refresh feature doesn't work either: when I end a turn, I always have to manually load the latest auto-save file. Does this work for you?
The CivAssist map, when I first ever switch to it, always gives me a blue screen - where the map should be - for a couple of seconds with a caption something like "Please wait, loading graphics". This must be quite an intense operation. I do notice that Civ III and CivAssist are very sluggish on this Win10 machine - much more than on my older (Win 7? XP?) machine. Windows-bloat...
But once CivAssist has loaded the map once, I can flip back to the Map tab, even later in the game with a now-different map, and the display is just there. One reason I can imagine you might be having problems is if your Civ install location is somewhere Windows considers "normal" (e.g. Program Files, My Documents, Users\[endless path]\something). It's been said way up thread that Windows does extra-annoying access shenanigans on those file-system locations - so I've got Civ installed on a completely different drive. But my theory here - about your particular problem - is completely speculative.
I haven't come across the "CivAssist suddenly doesn't understand a .SAV file as game time goes on" thing either. May be because I've only played 2 (unfinished) games on this particular setup.
If CivAssist doesn't ever auto-refresh onto the new .SAV, it must be running in "Manual" mode - look at the indicator in the extreme bottom Left: says "Manual mode", or something else like "On the lookout/watch". The latter means that CA has detected an instance of Civ3 (conquests.exe?) running, and is monitoring the Autosave folder for new files. I just thought:
if your CA is running in "Manual mode", then it may have difficulty knowing where the built-in game graphics are located; thinking it through, I think this is probably the only part of CA (apart from the Autosave folder monitoring) which "reaches out" to Civ's own files - the rest, once it's got the .SAV to start crunching it, looks to be internal code.
So it looks as though this "connection" from CA to the CIV files (graphics and .SAV files) is not being established in your case. When you run Civ or CA using your method, do you get that black screen with Windows' "OMG this executable might change files on your hard disk, are you sure?" dialog? That's a sure sign that they're being run as Admin. It could be that, using your less laborious method, they're not - and there's a guessable reason then why CA, as non-Admin, would not be able to sniff the other processes running to find Civ.
You could try my "shortcut set to run as Admin, then run the shortcut As Admin" method. It sounds crazy. But I do remember trying this Run as Admin using something like your method (set it on the executable only?), and it not working; "not working" in the sense that both Civ and CA do run, but they don't talk to each other. Really, running old software with Windows as it is, we're like druids speculating whether this year we should try doing the Special Dance anticlockwise, as that might please the Gods so that stuff works.
