Regarding 1), I would be certainly interested in what X can be shown to be true within Objectivism such that its establishment would undermine the system or cause a contradiction. As it is based on the Law of Identity I think one would be hard pressed to find an X that poses a problem, but hey, I'm all ears! :-)
Regarding the bridge between what-is --> what-ought. Certainly a hard problem prima facia. I'm not a trained philosopher, but I would say that, starting with "(A) != (!A)", one can say A "ought" to do action x if x preserves or enhances A's identity, that is to say, A's existence. So a black hole "ought" to accrete matter, because that is what black holes do, and accreting matter preserves and enhances the black hole's identity as a black hole, that is to say, it's existence, since, for Rand, identity is tantamount to existence. So man "ought" to be selfish because man IS by identity a selfish being. Acting selfishly preserves an individual's identity as a selfish being, i.e., preserves that individual's very existence. To "live for others" is to lose one's identity, and by declension, one's existence. "What-ought" it seems to me can have no meaning apart from the preservation of its subject or its actor. I "ought" to do that which preserves my genes, just like a black hole "ought" to accrue matter. There is no "ought" it seems to me divorced from preservation of existence, or preservation of identity. Thus:
1) Man is by identity selfish. 2) Man does not exist without his identity. 3) Man does not exist outside being selfish. 4) There is not "what-ought" beyond "what is needed to preserve existence of the actor, for all other meanings are subjective. 5) Man ought to preserve his existence. 6) By 2) existence is identity, so man "ought" to be selfish.
Admittedly a tough one, but that is my "stab" at it for the moment, ha!
Again, I would be interested in what X could be proved true within Objectivism such that Objectivism becomes inconsistent. :-)