How to finish Butt joint and corner bead within 5"?
Other than the first coat being forced into the joints, this is not two separate operations. You're going to be looking for one joint from the corner bead to the feather-edge of the butt-joint.

As for number of coats, as many as it takes, remembering not to waste time fussing over them - get mud on, let dry, sand or scrape for high spots, more mud for low spots, until there are no more low spots. There is no fixed number.