Question: If "Born in the Purple" is suppose to give my sons more right to be my heir, why is my first non-purple son still my heir? Is there a way to make my preferred son heir?
I sort of don't want to play CK2 until I know this answer. Please, someone has to have a clue.
I did a little digging. Born in the Purple affects inheritance for the Byzantine Empire, as does the Despot title; the two have equal priority, so granting an elder son not born in the purple the title Despot will rank them equivalently with the purpleborn. These do not affect lesser titles within the Byzantine Empire, however, and holding a kingdom specifically tends to affect succession in very "interesting" ways when you're Basileus kai Autokrator Rhomaion. As such, holding the Kingdom yourself is rather counter-indicated; the only reason you should form it under usual circumstances should be to give it away to keep your vassal limit in check. If your first heir is, as you mentioned, a republican burgher as well, that may affect inheritance for the Byzantine Empire title, which has some hard-coded interactions with Born in the Purple and Despot, but I have found no indication that it should prevent Republican inheritance. Unfortunately, without any of the information requested by umiman, I can't really say too much more.
An easy way to work it, though, seems to be to give away the Kingdom of Greece as a viceroyalty. As the Byzzies, you should have the laws in place regardless of the legalism requirements. In that manner, its inheritance by your eldest son won't strip you of your Greek demesne as it simply won't be inherited by your eldest son; it'll be held in trust by whomever you grant it to until they keel over, at which point it will be returned. If your eldest heir at this point isn't also Born to the Purple or a Despot, however, you'll get to do it again. You can also destroy the title if your prestige and vassal opinion can take the blow.
If you're absolutely intent on holding both the Empire and Kingdom, I won't stop you. It does, however, mean certain actions must be taken to get your eldest son out of the picture. It's been a long while since I've had to disinherit someone, but some more poking around suggests that you can do it by making them a bishop (not possible; already a burgher), forcing them to take vows, execute them, chop off their nadgers, or convince him to join a holy order (also unlikely due to being a ruler). I saw indications that a Doge also cannot inherit feudal titles, but I've also seen counterexamples that suggest that a Doge can and will do so, so that may not be a viable option.
EDIT:
Question: If "Born in the Purple" is suppose to give my sons more right to be my heir, why is my first non-purple son still my heir? Is there a way to make my preferred son heir?
I sort of don't want to play CK2 until I know this answer. Please, someone has to have a clue.
Pretty sure Born in the Purple only matters for the Empire. Lower titles won't care.
The purple son isn't inheriting anything at all, nor is he my heir. The last time I had a younger purple son we was my heir and was going to inherit everything. This game is so obscure I can't tell features from bugs.
He isn't inheriting anything at all? Not even the Byzantine Empire itself? Do you have any demesne lands outside the
de jure Kingdom of Greece?