1. If the player doesn't work with the Legion, Caesar would soon be dead from his tumour, or is killed by the upgraded securitron army rampaging through the Fort. Lanius can be talked down I suppose, but with him in charge I doubt the Legion would last long.
2. If the player sides with House/yes man, they would also more than likely be killed by the securitrons. Lanius is known to dislike Inculta too, and probably wouldn't make him his successor. You do have a point with Lucius however.
3. Well its not guaranteed, I see it as more than likely happening without Caesar. Lanius is a blood mad butcher, which doesn't exactly inspire loyalty.
1. Current Caesar is sure to die without the players intervention, or, if he do have a couple of months to live then it "might" be possible for them to find the last piece to the auto-doc or find someone with the medical know how to fix him. It's far-fetched, but possible.
When I upgraded Houses securitrons for his questline though they never showed up at The Fort, only at Legate's camp, so Lucius, Vulpes and comatose'd Caesar could sneak out if they have time.
2. Well, if Lanius becomes new Caesar then he might appoint a legionnaire we don't know of as the new Legate, doesn't have to be just Vulpes or Lucius, when the current Caesar dies the the power structure needs new commanders, 1 new Caesar, which is Lanius, 1 new legate, which is whoever Lanius appoints, and if it's Lucius then they will need a new leader of the praetorian guard.
Unless of course the new Caesar, be it Lanius, Lucius, Vulpes or Anthony, decides to change the power structure.
3. I dunno, the legionnaires are so brainwashed that they think of Caesar, who ever takes that title, is the son of Mars, so even if Lanius is coldblooded they would still follow him with "to the death" loyalty.
Infighting is another thing, if Caesar and Lanius both dies and it takes a long time to appoint a new Caesar then the morale in Legion will be low, but once a new Caesar is appointed no one (except commanders possibly) will argue against it, the new Caesar is the new god of the Legion and the rest will follow him (or her?).
The parts of Legion who have split off will probably either return or be hunted down anyway.