Likely FJ still blocks Rudolf from governmental responsibilities, and Rudolf becomes a sort of Friedrich III-figure. That said, he and his wife drifted apart due to their daughter's gender, plus he gave her syphilis as well. Here, they might stay together a bit longer, but I fear the result might be the same. If by some miracle, Rudolf manages to pry some sort of role from his father's fingers, he might not get so depressed that he shoots himself, but considering that he's 3/4 Wittelsbach (his mother being one and his father being half) it might be a case of
when rather than
if. (The Wittelsbachs of the last generations not exactly known for their mental stability - Ludwig II, Otto, Princess Alexandrine, etc)
As to the son, I don't really know. Sisi hated being a grandmother even more than she did being a mother, so chances are she still acts as OTL. Stephanie doesn't come off really well in Princess Windischgratz' views regarding her maternal duties. So, chances are this little archduke grows up with a rather disinterested family. Of course, FJ might display interest in him - but considering his behaviour with Rudolf didn't exactly win him 'father of the year' awards, Rudolf, is he has any interest in his son, might try to stop his father repeating the same pattern as he did with him. Now,
that scenario could be interesting, since Rudolf, was politically his mother's child, liberal-leaning, interested in federalizing the empire etc, which was part of the reason his father distrusted him (nothing personal, Rudi, he distrusted Maxie's liberal leanings as well (courtesy of the British who had proposed Max become king of seceded Hungary in the '50s).
If he's heavily influenced by FJ, who might view him as 'heir presumptive' first and then 'grandson', then I hope that he's got something of FJ's flair for governing. In the emperor's 70-year reign, he was basically the only thing that was holding the empire together. One book describes the Austro-Hungarian empire as "being like a Viennese pastry perched over mid-Europe - it looks pretty, but it's hollow".
As to Wilhelm II influencing him, that depends in what way. FF was friendly with Wilhelm II (IIRC Rudolf too), but are we talking the private pro-English kaiser or the bombastic militaristic public kaiser that influences him? Maybe he sees Wilhelm's militarization and longs for Austro-Hungarian imperial army and navy of his own. Not a bad thing, since same book as before says "Austria fought three wars in the 19th century, but never truly emerged from the 18th". So if it goes for military/naval reform, awesome. Wilhelm's political influence (FJ's too), has the potential for disaster. "A genius of statecraft might've welded these disparate realms together to form a whole, but the emperor [FJ] was no genius, and for the duration of his 70-year reign, he manfully held the empire together"
The kingdoms of the Austro-Hungarian empire are going to want some concessions - probably the same as were given to the Hungarians. There had been an attempt by the
Bohemians in 1871 which failed because it was blocked by the Austrian
and Hungarian elements in the cabinet. So maybe, somehow, Rudolf manages to get his father to see reason in allowing such a compromise. And we have an Austro-Hungaro-Bohemian empire come WWI