I think if you work this out on paper, you may find you are cutting it a bit fine for doing the relaxation thing. Let's give things a tentative walk-through.

Nov 23-24. Wellington. (I assume you have worked out the dateline change and that Nov 23 is the day you actually arrive in Wellington; remember that if coming from, say, Los Angeles, you would likely leave evening of Nov 21, skip Nov 22 as you cross the dateline, and arrive in Auckland on the 23rd to connect to your domestic Wellington flight.)

Nov 25. Most of day driving to Tongariro area. If memory serves, it's about five hours from Wellington. Plus, you will need to allow time to pick up car, if you have not done so already.

Nov 26. A full day for the Tongariro crossing, or if weather foul, some alternate short walks.

Nov 27 A half day driving to Rotorua. You might do a 'thermal' park en route, say wai-o-tapu. Or maybe you will want a full day in Rotorua.

Nov 28. Either stay in Rotorua, or leave Rotorua midday for Coromandel (sort of en route, whereas Bay of Islands isn't.)
Nov 29. Either leave Rotorua for Auckland, or stay in Coromandel.
Nov 30. drive to Auckland. Auckland.
Dec 1. You drop off the car, and leave.

As you can see, you have none too much time.

Not feasable to do both Coromandel and Bay of Islands, BOI is along drive up from Auckland and in my view not as interesting as the Coromandel Peninsula. It would be best to go there on the way up from Rotorua where you can turn off towards the Peninsula in Tirau but it's still quite a long drive even from Rotorua. Sounds a bit ambitious to me with your short length of stay. Have you researched the Tongariro Crossing carefully, it is a very strenuous hike, you must sign in at the ranger's headquarters and you will need to organise a pickup at the end of it. Are you experienced i n alpine conditions, it is a high altitude area and the weather can change rapidly from sun to sleet. Needs planning.