Do most Bombay cats typically have breathing problems?

NO - at least they should not. Even with a rounded head, they can breath normal. Bombays are basically "black" Burmese and should be bred to the standard.

BTW Persians have a much flatter face then any Bombays!

Yes, because of the way the Bombay nose is shaped. The top of their nose does not have a gradual slant where the noise is connected to the forehead. Their nose goes straight back to the forehead which comprises the flow of air to their lungs. In fact, that area of the nose goes further inwards leaving the air passage smaller.