Yes, but as I understand it, "cita" can have two meanings in Spanish. In this case, you are only talking about a romantic date ("go on a date").

Yes, it's perfectly phrased. Worth noting that this would only be correct if the proposal was to go somewhere.

It does make sense. "Go", obviously, implies travel away from where you are. But it also implies that your destination is some place where the other person would not normally be. If you are going to meet the other person, or to the other person's home or office, you might well use " I can't come on that date. "