New Zealand in early 2018

I don't have answers for all your questions but my couple of thoughts
Milford sound - stay 2 or more nights at Te Anau and drive from there. The drive in is lovely and driving in gives you the option of booking on a boat other than the lunchtime one that all the day trippers from Queenstown pile on to. We stayed one night in Milford so we could do the late afternoon cruise - that could also work.
North Island - Rotorua is pretty cool for thermal springs and Maori cultural shows. If you can spend some time in Lake Taupo - lovely natural springs running into creek. I can't remember exact directions but ask at your accommodation and they will direct you.
Most people head down the west coast of the South Island and see the glaciers. To be honest they were pretty underwhelming for me and if I were to redo it would go down east coast and end in Queenstown (pop up to Wanaka though, it is lovely).
3 weeks is the minimum to see both islands - don't underestimate the time it will take driving, don't expect to see everything in 3 weeks

Have fun. NZ is lovely.

One more thought after rereading your post - don't be put off by the prospect of rain at Milford - that is when you get the bonus waterfalls!!!!

Look at 8 destinations over 3 weeks.
Auckland - 3nts
Bay of Islands - 3nts
Rotorua - 3nts
Wellington - 2nts
Marlborough - 3nts
West Coast (Fox & Franz Josef) - 2nts
Queenstown - 3nts
Fiordland (Milford Sound) - 2nts

February is the peak month for tourism in New Zealand. When we went I made all of my reservations for accommodations in September to make sure that I got the rooms that I wanted.

Lord of the Rings areas are all over. There's Hobbiton on the North Island near Rotorua, which is also a good base for visiting thermal areas. I also recommend driving from Queenstown to Glenorchy and on to Paradise. It has gorgeous views. I'm sure that you can find LOTR maps on-line.

We went to NZ this past February. We stayed in Te Anau two nights, and really liked it. I echo the comment about the road to Milford Sound being lovely, and not to worry about the rain (dress for it). DO get sandfly repellent - they are annoying and itchy. We stayed in the Te Anau B&B and really liked it. I'd also recommend the national park visitor center and bird sanctuary.

February is their high season. Be sure you have your hotel reservations all lined up. We say "no vacancy" signs everywhere we went.

Lord of the Rings sites are on both islands. WETA studios are are Wellington and are worth a visit. Hobbitton is touristy, but really very cute. We took a full-day LOTR tour in Wellington, and I didn't think it was worth it. If you are a HUGE fan - maybe yes. Hobbiton can be reached from Auckland or Rotorua.

We flew into Christchurch when we first arrived (started on the south island) and took the Interisland ferry up to the north island (Wellington)

Dunedin was nice (opposite coast from Fjordlands). Good museums, Albatross Center, scenic train ride.

I haven't been to NZ yet, but I'm headed there next month and did a lot of research on Lord of the Rings (LOTR) sights before planning my trip. NZ ended up on my bucket list to begin with because of those movies, therefore it will be the primary focus of my trip. I initially planned on a fully DIY trip, but the LOTR locations are scattered throughout the north and south islands and some are fairly remote which made me nervous about getting to them all by myself since I'm horribly directionally challenged. Doing a quick google search I was able to find many companies offering LOTR themed tours, from day tours to Hobbiton from Auckland (which you can combine with other activities if interested) to horseback LOTR tours out of Queenstown, to a Hobbit themed "barrel run" kayaking tour on the Pelorus river, the possibilities are endless. Once you have an idea of where in NZ you want to go, I imagine it would be pretty easy to work in some LOTR activities here and there. I don't want to recommend any specific tours because I haven't gone on my trip yet, but I wanted to share that it was not difficult to find plenty of options whether you want DIY, day tours or multi week LOTR adventures. Good luck with your trip planning!

I've been to New Zealand a few times for work and holiday. I'm an Aussie. In my opinion the South Island is the better half. Christchurch use to be such a pretty place but a lot of the old quarter was still in tatters a couple years ago from the earth quake a few years before that. But it is still a pretty town.
But I fell in love with the Queenstown area. Big time. You could use this as a base and do day drives or tours from there, but theres so much to do in the town or just out.. Other towns in the area that are beautiful are Arrowtown, Glenorchy and Wanaka. A day or two in Wanaka could be an option too.
From Queenstown we did a couple tours with Real Journeys, those being an overnight cruise at Doubtful Sound which was fantastic and highly recommended, and then we did a day cruise at Milford Sound. And as one of the other people commented, if it rains in Milford Sound you do get the magnificent waterfalls.
Dunedin is also a very pretty town, which is across the east coast from Queenstown.
In Queenstown we stayed at Peppers Beacon. Can highly recommend that. But sorry I can't remember other places I have stayed at...