There are a lot of threads on here about what Franz Ferdinand would have done if he had not been assassinated, but there isn't much discussion about other could-have-been monarchs, so I thought I would make one.
What if Franz Ferdinand's son Maximilian was allowed to inherit the throne?
I'm no expert on dynastic succession rules, so I have no idea what kind of POD would be necessary to allow this. But what kind of effect would this have on the Great War? Maximilian would only be 14 in 1916, so I think he would need a regent. Who would that be?
Impossible. You'd need an entirely different Franz Joseph somehow allowing children from a morganatic marriage to be part of the succession. That would require such an intense and sudden break from tradition to make it happen. This is the late 19th and early 20th century, after all, and monarchistic traditions were highly important in the name of stability and legitimacy. Either way, since he was a minor, this would hardly affect WW1.
What if Karl Ludwig never got typhoid, and lived long enough to inherit the throne?
I don't know much about him, but he'd be 72 by 1916. That's pretty old. Otherwise, he was rather uninterested in politics and like Lalli said, he was a pretty devout catholic. Supposedly obsessively so. I think power would very quickly slip away from him, leaving power to his grandson and heir, OTL Emperor Charles.
What if Crown Prince Rudolph didn't commit suicide?
This, in my opinion, is the most interesting. Is his marriage different? Does he have children with his OTL wife somehow or with a different lady? Is he more mentally stable? If he does, then we have an absolutely and completely different succession than OTL. You'd need someonw creative to create a new succession starting from Rudolf's fictional son. Rudolf himself was more liberal than his father and often butted heads with him and his mother (despite being ideologically rather close; maybe it was due to both being mentally unstable). The other question is: would even Rudolf outlive his (clearly) semi-immortal father? Regardless, his impact will be before he succeeds him and the two will butt heads quite often. Plus, will Franz Ferdinand go down quietly and stay with thr navy or make this a three-way power struggle?
It's such an interesting and severely underrated PoD with so much potential.
What if Kaiser Karl died somehow, and Otto got the throne before the end of the war?
Like Lalli said, regency happens. Likely some kind of council that includes the Empress. The problem is that while she was relatively unknown during that time as she built up her reputation post-war during the family's exile, she had several rivals within the militarist and (German-Austrian) nationalist factions because she was an Italian princess whom hated her guts.
The problem with these PoDs is that they are pre-1900 and can make things very different if these characters make their impact before they take take the throne or they take the throne as OTL and at that point, there is nothing they can do to change the course of the Great War. You'd need more pre-1900 butterflies to change things other than 'Rudolf survives'. Hell, Rudolf might become the Black Hand's target if FF stays in the Navy (and doesn't do much politicking other than fight for more funding), which makes the PoD entirely irrelevant...