How to find out how big an opponent’s realm is

Open up your character window, and hover your mouse over your titles. An infobox should now appear, showing you the name of the title, the succession law and the line of succession.

I do not know a way to change the laws other than the one for your primary title, however. :/

EDIT:
Ok, I think I understand now. I believe it the core of the answer is whether or not a title is a de jure vassal to one of your greater titles. The succession law of a title will always be determined by the greatest title you hold in that "realm". If you have titles in other "realms", those will be determined separately.

• For example, if you are King of Ireland and the Duke of Munster (which is de jure part of Ireland), then the duchy will inherit the succession law of the Kingdom of Ireland.
• However, if you are also the Duke of East Anglia (de jure part of England), then that duchy will have its own succession laws.
• If you also hold the title of King of England, then the duchy of East Anglia will inherit its succession laws from England (which are separate from the succession laws of Ireland, even if you hold both titles).
• Now, if you also happen to be Emperor of Brittania, then I assume all the titles from within that realm inherit the succession laws from your primary title.
• Should you hold titles which are not de jure part of Brittania (say, King of Poland), then those will have their own laws, and the chain will start over from count to emperor, depending on your greatest title within that "realm".

PS: I know my use of the word 'realm' may be a bit different from the one used in game, but I could not find one that fit. What I mean by "realm", is a collection of titles which are de jure connected under one greater title which you control. I hope my intentions comes across despite this.

You kill your own Duke. So long as the Duke's son (whichever) inherits his father's lands before his mother's there is no way you lose Jazira. The chance of the Duchess dying first is why you are getting the inheritance warnings. If she dies first there's a chance that her heir could end up inheriting Jazira as well when the Duke dies. With Orkney having Gavelkind succession, if she has enough holdings there's a chance both of her sons could be landed vassals of another realm when your Duke dies. If both sons are vassals of another country when the Duke dies, you'll lose Jazira for sure.

As Affine and Studoku point out, the reason your High Crown Authority isn't preventing this is because Jazira is not De Jure part of Byzantium. It only can prevent out-of-realm inheritance in the De Jure Empire. I don't understand why Primogeniture succession in Jazira would be giving the younger son Jazira while the older gets Orkney, that's just backwards. However, so long as the Duke dies before the Duchess, and before either son is landed, I can assure you the inheriting son (whoever that may be) will be your vassal.

I know you have said you don't care to have Orkney, but if you did, the solution would be to kill the Duke, then immediately begin plotting to kill the non-inheriting son. This would assure that Orkney went to the son that just became your Duke.

As you say, this is Crusader Kings. All things are possible through adequate murder.

If for some reason you wanted to achieve your goal without killing anyone (a strange concept, but bear with me): If you have a title to spare (even a barony will do) you can land the son that is the heir to Jazira. By giving him the barony, he becomes your vassal. When he inherits further titles those will also become a part of your realm, even if the titles he inherits from outside the realm are of higher rank than his first title. So long as he is landed in your realm, any lands he inherits become and/or remain a part of your realm.