Painters tape will not be the best way to keep the mortar from falling through the bottom. if fact it has been my experience that painters take barely sticks to brick and concrete. It would be better to take a longer piece of wood from each side of the corner that you clamp to the underside of the brick coping. C clamps or similar adjustable clamps would work for this. Place the clamps away from the corner so that they do not interfere with your mortar project.
It would actually be ideal if you could arrange these pieces of wood such that they were mitered in the corner to aid in dealing with the corner pieces. The wood pieces could even have a curved molding installed that came up half way around the nose of the coping bricks. Something like this:
For placing the mortar between the bricks you may want to look into tools used by the tuck-pointing trade. There is a funnel shaped bag with a narrow spout that you can fill with wet mortar and then squeeze it out through the spout. Works very much similar to the gazing bag that a cake decorator would use. They are also called grout bags. Picture source is from a video that shows using a bag.
Cleaning up the mortar off the bricks is pretty easy if you use a damp cloth and wipe up any smear that gets on the bricks. I suspect that once you get a little practice with the tuckpointer's bag that there should be very little mortar to clean up.
Dealing with the corner pieces will be tricky. I might suggest bonding the two pieces to a piece of metal so that they are more of a unit and then bond that to the base with the construction adhesive. you could also consider looping some wire through a hole in the metal piece and then pulling the wire back along the bottom of the angle cut between the bricks where it would be embedded into the mortar.