I know I'm like 13 years late for this forum, but I found it, so others probably will too.
Civ III AI DOES CHEAT
I've been playing this game for something like 15 years. I have racked up over 3000 hours of game time. And this game cheats, cheats, god damn cheats. It knows maps, it knows resources before they drop, it knows what units you have in your cities from the very start of the game and all throughout the game (no espionage). Worse still, the original poster is correct. I have seen AI successfully kill a infantry with a level 1 warrior many times. I have seen my ARMIES of cavalry defended by two or three spear men. I have seen AI make a beeline for a city that was heavily defended for over 20 turns then just as I move my armies out, there they appear (no espionage). I've seen AI beat battleships with frigates. I've seen one AI destroyer take out a fleet of bombers. I've seen AI take out one of my cities with a single unit, only to have that city defended by that same unit and three or four defenders the next turn without any transports docking in, any airports, or any access at all. These units just appear. The game cheats.
I love this game. I obvious have some kind of obsession with it. But it cheats beyond doubt and beyond its allowances from about monarch level and up, BADLY! When I play at high level, I send in 40+ cavalry or tanks or whatever to take out one city. I have to.
The higher you go up in difficulty the more AI strategises against you. AI will research a different tech each, then share all techs with each other and refuse you. So for every tech you complete, AI is completing three or four depending on how many comp players are in the game.
I have always had this issue with the game. It is MASSIVELY, unfairly titled in favour of the AI. It always has been.