Are you allowed to tell Omega an integer that you can't actually calculate but that Omega can, if you know that integer exists and can uniquely define it even if you can't calculate it?

It doesn't really matter much. It changes the methods you use somewhat, but the basic result is the same. For that matter, you could even allow infinite numbers without doing a whole lot.

If you and Mrs. X both give the exact same response (for instance, if the above answer is valid and you both say it), then what happens?

I'd guess that you both get half a utility.