Tape has to go on the water pipes. Then the insulation. Otherwise any thermostat would be on too much if there is one. Plus on the outside of the insulation it pretty well defeats the purpose basically. Yes unfortunatly all the exposed copper pipes will have to be done basically.

You kind of mentioned the drain froze up as well? A five hundred dollar plumbing charge might indicate this. The local plumber you used should have made some recommendations as well at the time of the last repair. So if everything else is normal it might have been a drain trap that froze.

You should only have to keep the trap thawed.Depending on the plumbing arrangement you might be able to move it upstairs into the apartment. As long as the drains have some down slope they should not freeze. One quarter inch drop to the foot is ideal but you can get by with less. Just use a level on the drain pipes if visually you cannot see any downward slope towards the egress point.