Answer: It appears to be sufficient that the agent only knows that Omega is always correct. If this is the case, then a simulating-Omega and some-other-method-Omega are indistinguishable, so the agent can freely assume simulation.

This is false. If there is not a conscious simulation running, the agent will know he is not a simulation, and will two box.

Objection 2. The argument does not work for the problems where Omega is not always correct, but correct with, say, 90% probability.

As long as the probability is sufficiently high, and the agent is sufficiently uncertain of whether or not he is the simulation, it works fine.

Answer: The agent in the simulation has exactly the same experiences as the agent outside, so it is the same self, so it values the Omega-offered utilons the same.

If the agent is selfish, and his sense of identity is such that he doesn't consider the being he is a simulation of himself, the simulated self will not care about the non-simulated self.

I admit it does seem a bit weird to have a utility function that depends on something you have no way of knowing. It's not impossible, though.