1. Would the Savoys still reign had VE III abdicated in 1943 after the fall of Mussolini/Italy's surrender to the Allies? (or was the throne doomed at any rate because VE III acquiesced to the Fascists?)
Maybe yes, maybe no; Umberto II acted in a excellent way when in the period he was Lieutenant and later King, from 1944 to 1946, and for sure had Vittorio Emanuele abdicated as soon as possible after the armistice it would have been much better for the Royal House.
There were also plans for the abdication of both Vittorio Emanuele III and Umberto, in order to have the little Vittorio Emanuele (IV) as King with a Regent, but these plans failed because some wanted Maria José, who was well known for being very very liberal and with no ties to fascism, as Regent while some other didn't want the regent to be a member of the Royal Family.
Last but not least, we have to remember that in 1946 the republicans, in particular certain socialist and communist politicians, were very determined to topple the Monarchy, in a way or another.
So, I can't give you an accurate answer, just I'm persuaded that in 1943 the throne wasn't doomed at any rate.
2. Some have speculated that Umberto II was bisexual (and thus caused the separation). True or False?
As I have already explained some time ago, they were nothing more than rumors.
During the 1920s and 1930s the fascist secret police collected several files about members of the Royal Family and their relatives (Umberto, the Duke of Pistoia, the Duke of Bergamo, the Duke of Ancona, Prince Philipp of Hessen-Kassel) where were collected the so-called "evidences" of the homosexuality or bisexuality of the Princes (evidences like the friendship between Umberto and Luchino Visconti, noted homosexual, or even the fact that Umberto often gave to his guests as a present diamond broochs).
The aim of collecting these files was to discredit the Monarchy at the eyes of Italian people; Mussolini wanted to get rid of the Monarchy, and scandals about the homosexuality or bisexuality of the Royal Princes were a perfect instrument to blackmail the Monarchy. This is the origin of the rumors about Umberto's homosexuality.
Later, during the electoral campaign for the referendum, some politicians used these rumors for attacking the Monarchy and the King (the socialist Nenni used to charge the crowds up during election rallies with slogans like "Maybe do you want a pederast King?" or "if you don't vote for the republic you'll have a pederast as King").
Then it happened that during the exile the King lived separated from his wife and at the same time he didn't date other women, so some started wondering "maybe is it true that he is gay?"
They were nothing but rumors.
where is the truth? 
