Does anyone know if there's a way to view which technologies the AI Civs have researched, or at least how many? I know you can get information on whether or not they have any tech to steal from a spy (after setting up surveillance), that you can get a basic Tech comparison by mousing over the scores in the diplomacy overview, and that you get very infrequent advisor pop-ups about how many Techs each Civ has, but is there a way to see this whenever you want it? Thanks very much for your help!
Such information would be cheating in a strategy game unless you have earned that information (which the Civ5 only provides when the Spy has gathered the information). There has to be risk/reward in the game. The best way is to use your analysis of their units. The AI generally does a good job of having the latest and greatest.
Your trade routes, either active or potential, will tell you how many techs the civ that you are trading with has that you don't and vice versa.
The trade route info just tells you how many techs the Ai has, not which ones. For that information, you have to use clues garnered from the map and AI behavior. If you have a spy in their city, you can see what they have built or are building, which is always useful. Even without a spy, there are clues galore. Do you see their Cavalry running around on the map? They have Military Science. you see they've built (or are building) Forbidden Palace? They have Banking. Can you see the shadow of the Great Wall being built? They have Engineering. If they found the World Congress, they have Printing Press. Etc., etc.
You can also take cue from their Unique Units. Most AIs will beeline techs that unlock their UUs (and UBs). So, if you see Austria with Lancers (Metallurgy), it's a safe bet that she's aiming for Military Science to get her Hussars.
There are lots of other ways to gather clues, but that's part of the fun.