The reign of Gian Galeazzo is, I think, the most plausible POD for a united Italy in the Renaissance. However there are a few caveats. While Visconti would probably be best served to seek a crown from the Pope he would have to tread carefully. The second he becomes too powerful the Pope will just call in the Emperor or a King (France, Aragon, Hungary take your pick) or a powerful Duke (too many to count) to depose him and become the next King of Lombardy. After all thats what happened with Sicily and then Naples.
The reason I think Gian presents such a good chance is because Germany is in absolute chaos during this period. No ruler in Germany could consolidate power enough to come over the Alps and get crowned by the Archbishop of Milan and the Pope. The Visconti have time to build legitimacy while Germany is in chaos and the Papacy is divided. With a strengthened position in Italy, the Visconti could even end up taking an active role in ending the Western Schism.
Failing that the Pope could just create some league of Italian states, Venice, Savoy Naples whoever else is left and battle it out for control of the peninsula. He and his successors may be able to defend themselves but it will slow the unification of Italy.
Gian Visconti, with the taking of Florence, has broken the balance of power in northern Italy. By reducing Florence and Bologna he has established Milan as the head of a new northern Italian state, and the Papacy and Venetians are the only major powers who can threaten him. Basically, at the time of his death, he had beaten everyone, everywhere.
However a Papal crowned Visconti King in the north would legaly unite all of Italy under the Papacy such that even if it is a legal fiction (after all how much practical authority did the Pope ever have over Sicily) it at least eliminates the old Feudal networks of the HRE from the north. Furthermore a POD at the end of the Trecento as opposed to the Quattrocento pre-empts a few things, namely the Venetian Terra Firma, that would otherwise be threats to a united Italy. And if the Visconti can make a lot of progress in the early quattrocento then by the time the French and Spanish/Aragonese have a free hand to invade the new Visconti Kings may be too entrenched to be removed.
I'm not sure the Pope could break the HRE. I'm not really sure how it would work, removing one of the crowns from the HRE. I'm calling for Susano in this one (sorry about not using HREGN).
Venice already had started taking land, and would probably continue to do so. Conflict would definitely come between them, and in his the Visconti can look to the German Dukes in Tyrol and the Austrias, and to the Hungarians, for support against the Venetians. I think Venice will become stronger rather than weaker however, with the rise of the Medici Bank in Venice keeping Venice at the forefront of European economics.
A few things would likely never happen. There probably wouldn't be a Medici Doge. Sorry but the Medici have only just stepped into the world of banking at this time and are not yet dominant in Florence. Not to mention the fact that the Venetian Patriciate is a closed body. In fact most Florentines would probably head to Rome before Venice. Venice is a VERY different city from Florence. Assuming that Rome stays independent for a while I'd imagine that it would become the center for the Renaissance. If there's a Pope there banking will be big (it was the most profitable Medici branch) it has a great network of patronage (private, ecclesiastical, foreigners) and it has all of those great Roman ruins lying around for people to study.
Giovanni de Medici became the Papacy's favorite banker because in 1410 he bet the Papacy would return to Rome. The Papacy rewarded him. That is after the POD however, and because of the cosmopolitan nature of the Medici bank I expect that cosmopolitan Venice would be a better venue than feuding and bloody Rome.
I don't agree with you here. Florentines were merchants, and there was no better place for merchants than Venice. I am looking for the book now, but Venice abandoned Florence in OTL, hoping the city would fall because they expected Florence's weavers to come to Venice. Rome was dominated by feuding noble families who fought with the Papacy and each other. Venice is still in its Golden Age at this point, and I think it will be the most attractive place for Florentine refugees. The influx of refugees has the potential to shake up Venetian politics- the emergence of new wealthy outsiders is going to necessitate their inclusion in the ruling of Venice, and the influence of Renaissance thinkers during the same period could produce a different set of politics. Especially if the citizens' councils hadn't lost all their power yet, this injection could be what they need.