How can I keep my Kings from fighting each other?

I'm running a nice empire with, as any self respecting Emperor would require, absolute crown authority. The wiki says:

Vassals can no longer go to war (except to rebel) at all ... It also does not work for any king vassal unless they are your de jure vassal.

I have several King vassals who wage quite a lot of wars on each other, vassals, and outsiders. If I wanted to keep all of my Kings at constant peace, would I have to hold the de jure Empire title above them? E.g. Hispania for Leon, Castille, Navarra et al, Francia for France, Aquitane, Bhriotáin and Brugundy. So far, only the Holy Roman Empire and Byzantine Empire exist, so it feels very weird to have vassals beholden to crown laws under nonexistent empire titles.

Can Kingdoms ever assimilate into Empires? If this were to occur, would they finally be under my crown authority?

I noticed that I can set the Crown Authority for each of my Kingdom titles independently; I gather that my vassals have that authority as well. What would the impact be if they had a lower crown authority than mine? Their vassals would be able to declare war, yes? Would they be able to declare on just other vassals of their liege or would they be able to target my other vassals?

The simple way to limit vassal kings from going to war with your other vassals is to make sure that they have all of their de jure kingdom within their realm. Otherwise the AI will go about trying to achieve that by themselves. While in theory a vassal king can still try to press other sorts of claims on your other vassals, these are harder to come by. In practice the lack of any de jure claims seems to keep vassal kings from waging war within your empire.

Mind you I've never used absolute crown authority, as I prefer to leave option of declaring war on targets outside my realm open to my vassals. The law any self respecting emperor wants is imperial administration. That lets you revoke vassals from your vassals, even kings, without penalty if they're starting to get too powerful.

Kingdoms will drift into your de jure empire, just like duchies will drift into de jure kingdoms. All the counties of the the de jure kingdom need to be inside your realm, and the kingdom title needs to be either part of your of realm or not exist. It takes 100 years to complete. In practice it'll probably take longer because rebellions will temporarily remove counties from your realm, causing the timer to start going backwards.

I'm pretty sure your vassal kings aren't beholden to crown laws of non-existent empire titles. They should have their own crown laws if they're not your de jure vassal.

Crown laws only apply to de jure vassals of the particular crown title, with no respect to the de facto realm holder. This rule is exactly the same for dukedoms under kingdoms as it is for kingdoms under empires. What this means in practice is that in order to maintain absolute crown authority under all of your vassal kings, you need to either form the empire they are de jure vassals of and hold it, or make sure that title is not held by any other character and wait a hundred years for de jure drift to move the kingdom into your empire. Keep in mind that even if you hold the empire title that is the de jure liege of a kingdom, that empire title has to have absolute crown authority, at that point the crown authority of your primary empire title has no sway over the kingdom. I believe the same applies to Imperial Administration, but I haven't tested that (there's nothing to indicate otherwise, though).

Something to keep in mind is that laws apply to characters based on their primary title. If, for example, you control the Empire of Britannia, and one of your vassals owns both the Kingdom of Ireland and the Kingdom of Brittany (where the Kingdom of Ireland is your de jure vassal and the Kingdom of Brittany is not), the King will be subject to your crown laws if the Kingdom of Ireland is his primary title, but not if the Kingdom of Brittany is his primary title. Since absolute authority tends to cause a lot of rebellions, using this strategy probably is not generally a good idea, but can work if your current ruler is loved, respected or feared and the two kingdom titles will split after the current king's death, to give yourself some time to start incorporating the kingdom into your empire.