In response to some excellent questions from SonofWinter, this screen from a 1939 save shows my tech screen:
As you can see, I have 21 Leadership devoted to research. This means i have 21 projects running simultaneously, while another one is waiting until a slot opens up (i.e. when one tech is finished it goes to the bottom of the queu or I stop researching it for now).
This shows the theory tab from the same save. Here, you can research all the pure Theory stuff you want to increase your theoretical knowledge. However, since each tech takes several months to finish, and since research is 70% dependent on your practicals, common wisdom on the forum holds that it is a waste of time and leadership. The exception would be in you were interested in rushing nuclear technology, for instance. You could increase your nuclear theory by researching it here.
Your other question regarding sharing technology with your friends. There are two ways to do it. If you’re more advanced then they are, you can offer to Share Technology with them. In the following pic, i offer to share my Infantry Theory with italy, who is a member of the Axis. They will then be able to use my theoretical knowledge for their own research, but note the cost: 1 DP (diplomacy point) and 1 LS. The DP is no big deal, but the Leadership point is a quite steep cost as Leadership doesn’t exactly grow on trees. But if you wanted to, this is how you’d do it.
The other way around, to use the more advanced design of your allies, is to ask them if you could buy a production license from them. In the following pic, I’m asking Japan if I could build 1 Carrier Air Group based on their designs. Note that this will cost us 243 $ for 1 single Air Group, which we would still have to build using our own IC. Again, doable, but expensive. And no, as far as I know, no nation, no matter how good the relationship, will share their priceless capital ship blueprints with you. Unfortunate, but realistic. Otherwise, what would have stopped the Uk from simply buying the design for a Midway-class Carrier from the USA in real life? (a measure, I’m sure, from Paradox to counteract player abuse of the game system).
I hope this answers some of your questions.