Following that path yes it is likely but long-term survival is another matter. One major issue later Emperors would run into would be the divide between the two by geography, culture, and language and balancing the two. The Italian states, at least in the time of Barbarossa, were definitely the wealthier of the two chunks of the empire but they also were smaller and would have been less able to contribute substantial bodies of troops like the Germans did or be as agriculturally productive. It would take a pretty solid balancing act to keep the Empire in one piece to avoid being perceived as favoring one half too much at the expense of the other. The Angevin Empire, for example, was a pretty considerable power in its day and had a rock-solid administrative system but even with that it was still unable to keep it. It could definitely be possible for revolts or foreign invasion of Italy for the HRE to lose that territory if nothing else due to the fun of logistics across the Alps.
The balancing act you speak of would definitely be a major requirement in order to ensure the long-term survival and thriving of the Empire. It would be far from unmanageable, however. As you point out, Germany and Italy would end up being complementary halves of the HRE to a degree, and they may end up being recognized as such by HRE elites at large as the state progresses toward centralization. I would only add that the manpower and agricultural productivity of Italy, once properly harnessed by the end of feudal and city-state Balkanization, would be considerable, too, and the economic strongholds of a successful Hohenstaufen HRE would be several, in Germany and Italy alike: the trade centers of Flanders, northern Germany, Franconia and Palatinate, Bohemia, northern Italy, and Sicily. I would also add that a successful HRE would easily rely on the revitalized Imperial Roman/Carolingian mystique for building its cultural and political cohesion, and Latin would be an handy lingua franca. Heck, a successful HRE might just as likely lead to the revitalization of Latin as a spoken common language in the modern Empire, as much as becoming a giant Switzerland bilingual state.
As for losing Italy to rebels or foreign invaders, that could of course happen, if the HRE keeps having the short end of the stick as military talent and luck go, or a dynastic crisis strikes at the wrong moment. Just don't overestimate Alps logistic issues. Having particularist political instability on both sides of the Alps at the same time was much more the bane of the HRE Emperors than the logistic issues of moving troops through them.
As it concerns the Angevin Empire, I would say that much like the Ottonian and Hohenstaufen HRE, most of its failure was related to dynastic troubles happening at the wrong time, i.e. getting a couple of really sucky rulers in a critical moment for its evolution, at the turn of the 12th-13th century, rather than to its multinational character crippling it in an age when such issues were relatively minor anyway and its successful entrenchment since 12th-13th century might easily have sent European nationalism on a wholly different development path.
Not by the early 13th century they weren't, they had just beaten John Lackland under Philip Augustus and by the end of the Cathar Crusade were in control of all of France. By the mid or late 13th century France would be in a solid enough position that they could muck around with the Italian states or find other ways to cause trouble for the HRE instead of locking horns with England. While the English were a problem and constant thorn in France's side the threat of a strong, united HRE with the resources, population, and power given by holding Germany and Italy would have given the French enough incentive to make nice with England (which they did manage to do a few times during the 13th century) and find ways to upset the HRE's applecart before it gets too solid. There is also the question of cost for France vs. cost for Germany. If all the French are doing is loaning some money, selling weapons, and playing politics with the different local grandees (which wouldn't be out of the realm of possibility, France applied a similar policy with Scotland during Edward's invasion) that doesn't cost anywhere near as much as sending down armies to suppress rebellion. If the cost of keeping Italy exceeds the benefits the HRE might end up letting it go adrift and focus on its core in the north.
Again, of course all of this is quite possible in a best case for France, worst case for HRE scenario. If we reverse the terms, it is just as possible that an increasingly stable and successful HRE in early 13th century may spare some of its resources (quite possibly through the low-cost means you quote) to screw with France during the confrontation with the Angevins or the Cathar Crusade, so that the Capetingian monarchy enters mid-late 13th century considerably weaker than OTL. It would not be too difficult for a strong HRE to cast influence so that Lackland defeats or scores a draw with Philip Augustus, or so that Aragon, not Capetingian France, ends up the main winner of the Cathar Crusade.
Whether or not that happens, if France proves increasingly hostile to the HRE in mid-late 13th century, the HRE may be solid enough already to resist it, and it may react by forming an alliance with England and/or Aragon against France, an enemy coalition that France may find itself hard-pressed to counter, to the point of suffering crippling and irreversible damage in a Hundred Years War equivalent. A possible scenario that might develop from this could be a HRE-England-Aragon alliance facing a France-Castille-Scotland one.
Last but not least, let's not just fixate too much on Italy as the flashpoint of hostility between France and the successful HRE. Flanders, Burgundy, Italy, or any combo thereof, could be just as plausible flashpoints.
Yes they definitely could but it would have considerable long-term consequences. If, for example, the Pope manages to escape one jump ahead of Imperial troops he could continue to cause trouble threatening excommunication, interdicts, and all those other nasty tricks the Papacy used for dealing with recalcitrant monarchs. The Pope might not have much political power but he does have a lot of spiritual and moral authority that can prove problematic to any ambitious European monarch. If you have a weaker Emperor this could end with capitulation, a stronger one and you could see the Great Schism happening early.
The spiritual and moral authority of the Pope would carry real teeth against the Emperor only as long as it can rely on particularist opposition to Imperial rule in Germany and Italy being strong and successful. If the Emperor has sufficient backing of the German-Italian elites, but an hostile Pope escapes capture and forcible removal, he could and would easily counter excommunication and interdicts by raising up an Antipope, and one with the backing of the HRE at large would carry so much authority and influence over the Church that a true major schism would indeed emerge, if the Pope may get some serious support by some other major European monarchies in turn. So indeed with a strong HRE an early Great Schism is a very likely possiblity during the power struggles with the Popes.
Of course, the religious feud would become even more entwined with the secular power rivalry between the major European monarchies over time. In such a TL, unless either the HRE bloc or the French bloc win a quick and decisive victory in their struggle, I foresee the entrenchment of the *Great Schism and its ultimate outcome in the effective destruction of the Papacy and the Church evolving in a much more decentralized structure and politically cowed attitude, with most ordinary authority in religious issues being in the hand of the various national episcopal councils (under the watchful oversight of the local monarchs of course) and occasional ecumenic councils being the ultimate authority over the Church at large. The Church would thus evolve on a path much resembling the Orthodox and/or the Anglicans. Quite likely, this would butterfly away the Reform and/or reshape it to resemble the Anglican model at large, the various kings and emperors pushing religious reform to entrench even more secular supremacy and decentralization on national churches. It could also lead to an healing of the Eastern Schism, since Papal authority always was the main cause of the Latin-Greek split and stumbling block to a reconciliation.