The CivAssist program opens, but the window doesn't show when the game is running.
Does anyone have any suggestion for this problem?
On a single monitor, even if CAII is refreshing itself at every (auto-)save, the main screen may still stay 'hidden' behind the game.
Can you see the little 'Swordsman' button in the top-left of your game-screen (i.e. below the 3 game-standard menu buttons: Main menu, Civilopedia, Advisors)? When you want to consult CAII, clicking on that button should bring the main screen to the front (or pop it back up if you minimised it).
Or you could activate 'Show alerts', which should cause both the Alerts window and the main screen to pop up at the beginning of each turn.
There is also setting in the Options menu along the lines of 'Always keep CAII on top of other applications', which might help.
(I run dual-monitors on my Win8.1 desktop, so while I'm playing Civ3 on the primary, I usually just keep CAII — and the Alerts window — open on the secondary; when I play on the old WinXP laptop though, I sometimes forget to switch on 'Show alerts' for the first couple of turns of a new play-session, so even though CAII is running, I miss the updates until I remember to fix that).