Can this messages be turned off?
Some but not all. If you go to Preferences (Ctrl-P), you can turn various of the pop-ups on and off, according to what you want to see/how much you want the game to decide for you:
Activating "Capital governor is default for new cities" might not be the best idea, since (1) the Civ3 AI is fairly dumb and (2) a new city usually has different priorities than an established city.
Deactivating the "Wonder-initiation" and "Civil disorder" pop-ups means you won't get any warning about those things, so you
might overlook a fixable happiness problem (though the flames/smoke should also give you a clue!) or fail to speed up a Wonder-build. (Not sure if the "Civil disorder pop-ups"
also includes the WLTKD announcements; but in that case the fireworks indicate the celebrations).
Deactivating "Ask for build-orders..." (and activating "Always build previously built unit") means that the city governor may/will decide what any city might build next (if the Governor is set to 'Manage production' in 'All cities' then this is no longer a 'maybe'). But as I say, the Civ3 AI is fairly dumb, so generally it's not a good idea to let it play the game for you.
"Disable Population limit warnings" means you won't be informed that a town needs an Aqueduct or Hospital to continue growing. (Might also mean you don't get warned that a town has starved, not sure).