By no Mayerling Incident do you mean it does not happen or does not succeed? There are competing theories about it. The recognized theory is it was a joint suicide. The less popular theories is it was an assassination by German, French, or Austrian Intelligence personnel. This is based on the fact Baroness Marie Vetsera was battered to death (not a way one normally commits a suicide pact), and Rudolf used a gun he did not own (guess he had to buy a new gun just for this) to shot himself six times (must of been a really bad shot at point blank range).
The who is a big if. Rudolf was a well known anti-German, so it might of been the German's who killed him to prevent Austria from breaking from the Alliance with Germany and allying with France or someone else. France might of killed him if he backed out of "a deal" with them, and to destabilize the Austrian monarchy. Lastly, Rudolf was a known liberal and frankly was probably considered very weak by the powers that be, so the Austrian's might of removed him themselves. Of the above three I view Germany doing it the most likely, then France, and Austria killing one of their own the least likely.
Now a living Rudolf:
He was more liberal than his father, and was known to not like Germany or an Alliance with Germany. However, Franz Joseph did live to 1914, so as Rudolf grew into his role his views might of changed or hardened even more. To be honest Rudolf always struck me as a weak willed person, so you might have a situation where a powerful group of Ministers actually run things. This could be better or worse as I think the Hungarians would walk all over Rudolf, which might rally over one to him (to fight the Hungarians) or split the nation apart. However, this all assumes Mayerling does not happen, if Mayerling is an assassination and does not succeed then Austria will quickly develop an anti-German or anti-French stance, not to mention the huge international embarrassment that would be for the plotting nation.
Sissi died not last long after her son, so her life might of been better. Though, I doubt it she was a flighty flake. Though, the Brits and the Hungarians liked her, so she always gets good treatment in English language histories.
Rudolf's wife could not have children cause he infected her with a STD. Rudolf liked going to brothel’s. Though, supposedly his wife (Stephanie) was not very faithful either. So, Franz Ferdinand or his children (assuming he married differently) might still be in line for the throne.