I feel that I have a decent grasp on nearly every aspect of CK2, but I'm hoping that someone can answer the last few questions I still have.
1) Is there anything I can do about titles that are about to pass out of my realm? I can't think of a way to solve this without revoking the title, which I obviously don't want to do.
2) When I first go to war with another country, I'll begin getting these popups that say I just lost a battle or siege, even though I wasn't involved with one, and I'm not sure why. Sometimes I'm down 20 or 30% before I even engage in major action.
3) I instituted medium crown authority, but vassals are still going to war with each other inside my realm. I'm guessing that they can still rebel against their liege. In that case, how can I change succession laws when they are perpetually at war with each other?
4) I suspect I already know the answer, but I don't suppose that there is some way to bring another emperor within my realm. I got a claim on Byzantium and wanted to give it to my heir, but I don't want to give him any sort of autonomy, just so that the AI doesn't mess anything up.
It was quite a glorious curbstomping, but I'm having issues keeping my vassals happy. They always seem to come to hate me no matter what I do. Is it the fact that I'm holding several dukes in prison for plotting treason? Or that I'm basically setting up my heir to directly own a third of non-Muslim Iberia?
As Clevinger said, you can hover over the number to find more information about what is motivating their opinions. In general, there are a few reasons why your relationships might be suffering. 1) You might have something they want, such as a county or duchy. Similarly, holding too many duchy titles will affect the number. 2) Acting in a tyrannical or generally dishonorable way. This can include executing prisoners or dishonoring alliances. 3) Personality, religious, or cultural differences. These are difficult to avoid. 4) Your crown authority can result in a heavy penalty if it's too oppressive. Also, changing succession laws may polarize your vassals' opinions.
Consequently, there are ways to improve your relationships with your vassals, such as giving them the titles they want and occasionally releasing prisoners. Crushing a major revolt will also give you +25. Sometimes it's a good idea to attempt to imprison someone who is proven to be involved in a plot (especially if he's involved in a particularly subversive faction against you). If he rebels, then you can crush him, take one of his titles without incurring a penalty, and impress your vassals all at the same time. Another important facet to remember is that your relations will naturally improve over time, because vassals are impressed by high prestige and long-lived rulers. In my opinion, it's rare to experience too much seditious behavior beyond ten years of rule, assuming that everything is well-managed (for example, keep everyone happy and try to avoid long, exhaustive wars that will test your vassal's patience, as long as it can be helped).