Irredenta (Nice, Savoy, Corsica, Tunisia) were in French hands too.
France was a historical (in the modern sense) ally though, and a much stronger opponent than the teetering Austro-Hungarian empire.
Sorry, no. Italy was never that kind of psychotic about the irredenta. We were never going to love Austria, but if the best way to affirm Italy's status as a great power was an opportunistic alliance with Vienna, so be it.
The Italian people, or rather a significant section of the populace, were in fact die-hard nationalists. The final defeat of the Hapsburgs and the end of the
irredenta were a major part of late 19th century/early 20th century Italian politics, you can't deny this.
Sorry, no. Italy expected to fight France in a general war during the 1880s-1890s. Crispi was a big supporter of the Triple Alliance, and certainly it didn't harm his stay in power any significantly.
The Italians also expected that they could confirm their status as a European great power by taking Libya and Ethiopia, and look how that turned out.
Don't forget Austria was bound by treaty to give compensations to Italy ever since it annexed Bosnia, more so since it started the conflict by attacking Serbia, and Vienna stubbornly refused.
Anyway, this is beside the point. The main foreign-policy concern of Belle Epoque Italy is to affirm its great-power status, more so if it can do so by cutting down one or another of its traditional rivals, Austria and France. They are going to do so from an opportunist stance, and an Anglo-German alliance is surely the superior one. To recover the irredenta is important, but nowhere so much they are going to pick the weaker side, and besides, they got them on either side.
All the more reason for Rome to spurn a close relationship with Vienna. Neither party truly believed in the Triple Alliance, it was only German conservatives who flocked to Bismarck's ideal of 'containing' France that believed it would ever truly work. Remember that the alliance only came about after the League of the Three Emperors fell apart; indeed the Germans, both in Germany and Austria-Hungary, cared not a twit for Italy, they were only interested in having a useful pawn to gather leverage against France - and the Italians knew it.
The supposed Anglo-German-Hapsburg one is definitely the weaker choice for the Italians. They'll be fighting the French along the mostly open Franco-Italian border and in the Tyrrhenian Sea, the Central Mediterranean, which is going to be dominated by the French navy, and, depending on the time of the war, also in North Africa, where the French will certainly win out; where as a fight against the Austrias, as in OTL, will see fighting from in the mountainous Austro-Italian border over Trento, and in the Adriatic, which is fairly easy to bottle up, especially considering the Austrians only viable access to the water is Trieste, and possibly Fiume, though again this will be easy for Rome to counter. Beside all of that though, Vienna is clearly the weaker target, militarily, economically, socially, and geographically, the one that is the historic enemy, therefore the one that the Italian people will be more enthusiastic about fighting, and the one that has more defined and guaranteed goals and rewards with much smaller risks and investments. If Italy joins any side in an European alliance system, especially one with France and Germany as rivals, she's going to be on the bleeding edge of the fighting, but at least aligned with France (and Russia, and potentially Turkey going by your scenario), she'll be in the side that offers her more advantages and the potential for greater gains. In short from a strategic and a tactical point of view fighting the Austrians simply makes more sense.
I know we've had this discussion before Eurofed, but honestly I just don't see your much vaunted scenario of a reborn & greater HRE via an Anglo-German-Austrian-Italian alliance taking place without a POD at least as far back as the 1830s. After 1848 there simply isn't really much chance at all for all the pieces to fall into place to get all four powers to align just right, let alone the other states you intended for them to dominate.