I'm not quite sure where you're going with this. The BB numbers are definitely regular integers in the same way that twelve or a googol is, but the uncomputability and fast growth of the function tend to "infect" other transformations and related functions, so that it also becomes very hard to say specific things about them.

Yeah I don't really know either. Something like if you name a number by saying BB(10000), but it's not computable, then have you actually named a number? But really that's just a boring question of how you referee the game.

Of course. The number of Turing machines is countable, so you can number them. HALT(n) is 1 when Turing machine halts and 0 when it doesn't.

Incomputable but always 0 or 1.