Did I understand the status of Rome correctly - while the rest of the Papal States became part of Italy, Rome is technically a separate country? As you mentioned the parallel of San Marino - IOTL, its representation in foreign policy is handled by Italy. Is it the same with Rome? Is there the same separation between Holy See (the Pope as sovereign) and territorial state (Vatican City resp. Rome) ITTL?
The answer to all these questions is yes.
What happened to all those Italian princes, especially those that weren't Habsburgs?
The heads of Parma, Modena, Lucca et cetera have fled to Spain, Sicily or Austria. A lot of the minor nobility, however, joined the rebellion and have landed on their feet. (The Cavours, for example, are doing fine.)
According to the scraps of information I've managed to find online, there was no question about Hungerford producing heirs; he was gay:
Congratulations, you've made the 500th post! Any idea what Hungerford died of?
(Even if the U.S. doesn't get the money, there's hope. J.Q. Adams was a great promoter of education and science IOTL. For example, his first message to Congress, he lamented that there were more than 130 observatories in Europe and not one in America. He called them "lighthouses of the skies," which everybody in Congress thought was hilarious. ITTL, he may find a way to set a little money aside for the pursuit of knowledge.)
I consider myself a fairly good writer; I'd be happy to help craft "excerpts" from Italy Reborn.
Since Italy Reborn is supposed to be a collaborative work, I might even take you up on that.