Just out of morbid curiosity, what might happen if a Holy/Eastern Roman Emperor tried to take the title of Pope of Rome/Patriarch of Constantinople for themselves and merge it/them with the title of emperor?
No.
I suppose you want more, huh? Alright: The West had a strong tradition that the Church was independent of the temporal rulers of Christendom going back all the way to Ambrose.
What of the Jus Exclusivæ, last exercised in 1903?
Depends on how he goes about it.
If he claims he's the reincarnation of Jesus he can say that he is God's representative on earth and therefore the Papacy/Patriarchy should be his.
He could go ultra divine will.
He could found a new heresy in which he is supposed to be Emperor and Spiritual Leader.
A caliph could conquer the ERE and Patriachy, but that's cheating
Etc.
And if the emperor doesn't produce an heir then fuck civil war at:
Boy, this emperor is really angling to be assassinated and/or deal with a major uprising, eh?
Hypothetic Emperor Z or HRE have three sons
First son is Prince A, original heir to the crown, but he's a known vicious tyrant who clearly evil, but he's the lawful candidate. He loves to rape women and mutilate children.
Second son is Prince B, better military commander and such, as well as having some friends in the military to pull something off, but he's also known for his love to raiding and kill men without any reasons
Third and youngest son is Prince C, a good, kind hearted young man who is also very religious. Since Emperor Z doesn't want him to be corrupted as his brothers are, Prince C was sent to seminary to get a career in clergy.
And so everything goes well, for several years, and Prince C is now a Cardinal (partially thanks to his piety and such, but also his connection), and before long, Prince C is elected as Pope in fair election (due to his spiritual inspirations and such)
Emperor Z then get assassinated, but it was unclear on who sent the assassin, either Prince A or Prince B could be the one who done that.
Civil war erupts and long story short, both Prince A and Prince B end up dead in battle, for drama's sake, they kill each others in a duel...
Pope C is now the legal claimant of the Throne, and he decides to press his claim...
No.
I suppose you want more, huh? Alright: The West had a strong tradition that the Church was independent of the temporal rulers of Christendom going back all the way to Ambrose. The East was much more accepting of the Church being integrated in the government, but much less accepting of the clergy being involved in things like warfare.
So, no. He'd be declared mad and deposed if he was lucky.
Completely different scenario than what the OP was suggesting. Being a cleric doesn't invalidate one to inheriting a throne. Look at Cardinal-King Henry of Portugal. However, you forget that the Holy Roman Empire is an elective monarchy. The new Pope can claim whatever he wants, but chances are the Prince-Electors would chose a new dynasty/ a distant relative. The most the Pope could do in such a scenario is cause a civil war with the new King of the Romans/Emperor.