1) Who would be the other claimants? If there is a sufficiently prestigious Roman, then he will be the most likely to get the throne, the Romans liked having a Roman on the throne, it would be exceptional to have a German on the Throne of Caesars.
2) A successful claim on the ERE could lead to Rome becoming an Administrative Capital (perhaps somewhere in Germany would also, Aachen?) and I'd wager he'd try and rule from Constantinople, and specifically set up a German quarter to strengthen the German presence in the city.
An alternative would be to rotate between capitals. Aachen (for the Germans), Rome (for the Italians), and Constatinople (for the Romans). Though I expect the former to be much more likely.
3) By this point, I think people have learnt that co-emperors are dangerous, especially over so much territory. Setting up lesser Kings/Despots, or Governors would be wiser. With his inense interest in establishing new administrations, you could well see the various Duchies of his former territories competing to have the title Viceroy of Germany, or similar - it is essentially Kingship till death.
4) Great Schism is butterflied thankfully - would it become Latin? Greek? Coptic? who knows, beyond some talented theologian. I'd wager that for stability the Latin Pope wouldn't be too defanged, but instead given Primacy in the West, with Constantinople having Primacy in the East. Alexandria still remains a question, as does the Nestorians in the Far East.
5) Does it stick? Depends on the quality of his rule and his heirs. Strong Emperors that can play the church by conquering heathen territories? Probably. Allowing some decentralization of the Church between the two major Pentarchs allows everyone to enjoy their beliefs, but less orthodoxy - you'll need to try and encourage the church to have Universal Doctrine, under which there are acceptable Heresies.
6) German Roman Emperors - to be fair, it depends on when they were born. Those heirs born before Otto takes the throne are likely to identify as German. Those after as Romans (who are also German). This is probably where the inheritance of Germany and the Empire leads to problems. If there are only Roman Heirs, the ERE is going to be calmer, if only German heirs, the rest of the Empire may remain more loyal - but again, there needs to be a powerbase for Otto and his heirs near Constantinople in case the Romans consider them 'too German'