So many questions. So you got the Great Library and now what? First, let's review the fact that you can milk this for a while, but once two civs learn education, so will you. Then the "GLib" has no more benefit to you aside from generating culture. Some people actually give it away before they learn education so that they can re-capture it later for even more techs, but that's gamey if you ask me.
For the trades, if the Byzantines have the most gold, and it's the best they can offer I would say it's not a bad deal. Now once you trade it to them, they will turn around and sell the tech to other civs who can pay for the tech. That way it reduces their own expense. So it's a good idea to trade the same tech around when you can, understanding that 10 gold and such would be a case where you would probably not trade it.
Regarding government switches, at emperor level, IMO you can get away generally with changing governments twice where you are non-religious. Levels above that and once is enough because the tech pace is that much faster.
Feudalism and monarchy. You don't hear too many people touting the benefits of feudalism. Yes, you get MP bonuses, but war weariness can kick in over time as your feudal lords and barons get upset that their serfs are stuck in your armies and not in their fields. And science levels are similar to that of monarchy. So this government seldom jumps out at you as a preference. Monarchy is for war. You get MPs and no war weariness. But the republics will generate more science, and those who like to build their civs peacefully do fine with it. So if you want war, go monarchy and conquer lands and luxuries and cities that eventually you can turn into more production or research centers with scientist specialists. You want better science and follow a peaceful builder strategy, you can afford to wait for republic. There should be no rush here, you will get it for free eventually, and meantime at zero science you should be collecting nice stacks of gold each turn. Again if you want to switch later you can, but at that point it would be best to remain committed to that form of government until the game is over. The other government choices that appear later: democracy, communism, fascism are interesting when playing themes or variants, otherwise you can really do quite well simply with the basic monarchy or republic choices.
Now some players war successfully with Republic too, moving the luxury slider as high as necessary to keep the population in line. Given this is your first attempt at emperor, you may want to just follow the simpler strategy.
Now with the Incas, not sure where they are, but they're likely going to have high culture. So if you take Cuzco, you have to be prepared for a culture flip, or take them out entirely to prevent that risk.