1) There is no way to find out exactly what it is researching, but you can find how the AI values different things in a tech
here. Some techs that the AI seems to not research right away are Currency, Code of Laws, Printing Press, Medicine, Scientific Method, and other ones that don't give any benefits like wonders/improvements/units, etc.
2) Yes, the AIs very much like to trade between themselves. In fact, there is an AI to AI trade rate for difficulty levels. The higher the level, the higher the rate is, so the more they trade between themselves. On Monarch, the rate is 150 (Chieftan is 110).
3) Anytime you have a monopoly on a tech (you are the only one with it), and you sell it to an AI, they will most likely sell it between themselves the next turn, so sell it for whatever you can get if you decide to trade it to one civ. Military Tradition only gives Cavalry, so if you aren't planning any wars with them (or are waiting for Infantry, Artillery, and Tanks), hold onto Steam Power, as being the first one to have railroads can be important in building improvements and wonders, and it helps in wars too.
Good luck on your Monarch game.
