2-3 weeks for both islands - is this plausible?

Much more unlikely if hitch hiking. And freedom camping is something that is becoming much harder to do IF you dont have a self contained vehicle.

Ah ok.

Meaning I'd have to sleep in a camper van, or just a car that I can drive from spot to spot with?

By fully contained, I mean toilets, holding tanks for grey water. In other words, you leave nothing behind. You could sleep in a car in some camp grounds..

It would be a rush to go to both Islands in this amount of time. The South Island would be the best Island for your wish list, the landscapes are fantastic and you are more likely to see the wildlife there. Swimming with seals (and even dolphins if you like) is an amazing experience in Kaikoura.
The LOTR's locations are across both Islands. If you do go to the North then the Tongariro Crossing is a must, as is the Putangirua Pinnacles. I was blown away by both of them.
Enjoy your trip, it really is a magical country :-)

Well, what if my only real priorities in the North Island are one or two key LOTR locations (Matamata above all) and the Glow Worm Caves? Those are pretty touristy spots I'd imagine, so I doubt I'd have trouble getting to them (I'll rent a car).

I don't know when I'll be back, and those were big highlights for me, plus the other places you mentioned might be worked into my drive down anyway.

I looked at some typical 'two week' itineraries for both islands, and both include several places I don't really need to see. If I was able to devote two full weeks to the South Island and maybe 5-7 days to the North Island, how would that look?

Matamata's not so touristy. Its just another rural town these days.

Your itinerary would work but it would be a rush. Matamata and the glow caves would take 2 days to fully enjoy, Tongariro crossing a full day (and a must do), then the drive to the Putangirua Pinnacles another day. Two weeks in the SI is do-able depending on what you want to do, but be aware driving distances there take a lot longer than what the maps usually say.

Oh, BTW, there is a seal colony close to the Pinnacles. It's a beautiful wild coastline and well worth spending the day there

If you rent a car you will be able to drive to isolated beaches & National Park campsites AND carry camping/ cooking gear & food - all much harder to achieve by hitch-hiking.
Re the typical 2 week itineraries that go to places you don'twant to go - sometimes they are necessary overnight stops to break a long journey. In a 2 week circuit of the SI, typically starting from the Picton ferry, west to Nelson, down the west coast, across to Wanaka/Queenstown/ Te Anau & back north via Mt Cook to Christchurch I can't think of any overnight stop that isn't scenic & worth visiting.