For me, a Post-1204 scenario is contingent on the right leader emerging at the right time, and that is by no means guaranteed. However, lets assume for the sake of argument that we do. A reasonably competent dynasty has control of the Empire at some of its greatest height, and has an interesting opportunities on its doorstep.
1) The Mongol Conquests
a) As a result, the disintegration of the Sultanate of Rum, allowing the Romans to rapidly take advantage of its collapse. This is great for the Romans as they can engage in a rapid reconquest. However, as highlighted, the Mongols need to be kept on side.
b) Mongol Alliances - much like Russia in the north, the Romans could well take a position of an enforcer, or extension of Mongol Rule, working with the Mongols as allies whilst paying tribute to restore large swathes of the Roman Empire, because that tributary can project Mongol power. This means a SIGNIFICANT Mongol influence, and potentially a Romano-Borjiginid dynasty eventually takes the reigns of the Roman Empire.
c) Golden Horde-Ilkhanate Feud. Even if (b) doesn't happen, a stable Roman Empire between those two powers is obviously one that could be the focus of diplomatic efforts to destroy the other. That could bear fruit.
2) Lombard League - this is a huge deal, and could go a number of ways if the Romans are able to apply influence.
a) Roman support - the Lombard League as an ally of the Romans opens the door to the Romans re-entering S.Italy, and would potentially lead to the Lombard League becoming permenant (they want neither their new ally, nor the HRE as their rulers)
b) HRE support - the Romans could agree with the HRE to split Italy, and mutually apply authority over N.Italy. For the HRE, it means they don't need to repeatedly come south, for the Romans it puts them back in Italy. The key bit is how the power-sharing works. Do they both have a split of tribute? That I'm not 100% clear on. Alternatively, in exchange for 10% of the revenues of N.Italy in perpetuity, the HRE greenlights the Romans dealing with the Italians and annexing the region outright. The HRE suddenly has quite a lot of gold coming in for next to no effort. (And the Romans have Italy, and the Pope, in hand).
c) Papal Alliance - the HRE and Roman Empire are both able to counterweight each other. The Papacy could benefit from having the Romans in S.Italy, but the HRE in N.Italy once again. It guarantees Papal independence.
3) Reform. Again, the Mongols are so important for this period. The Mongol Empire uses similar strategies to some periods of the Roman army - mobile ranged cavalry, support troops, etc. This doesn't mean that both sides can't learn from each other. As stated earlier, I can see a Mongol-descendent family rising to the Purple. But the Mongols also open the door to ideas flowing east and west aggressively, and provide a period of peace, and likely brutal enforcement of that peace, to minimise civil wars in the Roman Empire. Long enough to instigate reforms. We could see all sorts of ideas come from China, or the Mongols. Potentially the Romans institute a system of Exams like in China, or take on board the Mongol merit system. Plus, with the Mongols enforcing peace on the Asian borders - a lot of the issues the Roman Emperors had with border Strategoi becoming very powerful and popular dissipates as they aren't needed. That border can be brought back under direct control, or even the Theme System abolished in favour of a central administration and Tagmata once again.
4) Expansion. I mentioned in 1b, that the Romans could be an enforcer and an extension of Mongol rule. This could facilitate a truly frighteningly large expansion. Romano-Mongol campaigns could hit Italy, Hungary, Africa, Arabia, Syria. We could see the Mongols establish a dozen smaller states that the Romans are in charge of collecting the tribute for, and sending it to Karakorum. The Romans also have to enforce that rule. Which means that if/when the Mongols collapse (I mean, the Romans could prevent that, see 1c as a potential root cause there) the Romans can pull a Moscow and walk into their former "enforcement zone" and establish either direct rule, or a series of eventually integrated Katepanates/administrations.
This is DEEPLY optimistic, because the alternative is that the Roman Empire is promenent enough to be seen as a worthwhile target. This could mean direct attack, or the Mongols partnering with the Seljuks to establish THEM as an enforcer.
EDIT : I realise I've completely missed out dealing with Bulgaria - that is an obvious potential, again on the same grounds as dealing with the Seljuks. Mongols smash, Romans waltz.