>What are the temps like in NZ during the day and night? I've heard there's lots of rain which isn't a big deal as long as I have the gear for it.
I had a few cold and wet days right in January at about 8 C: chilly if it's raining and windy and you're dressed for summer. And no doubt it can be colder, it can still snow in January if you're unlucky. So I would give yourself a bit of margin, maybe a fleece vest/top which you can wear at night anyway. Max for me was 28 C, but it's a fairly pleasant dry heat, and often you're either up high where it's cooler and breezier, or down low in the bush out of the sun.
Re gaiters: I only wear them for rough off trail bushwhacking or when in snow (even then I get away with just my rain pants a lot of the time). On the trails or above the bushline (90% of my walking) I don't wear them. So it depends on personal preference and where you are going, don't feel they're mandatory.
>My idea was that this shoe could double as a river crossing shoe as well so that my boots could stay dry!
It's unlikely your boots will stay dry, even if it isn't raining. Ground is often boggy/swampy/muddy.
>I think I'll stick to the huts instead of camping out so that I don't have to bring a tent.
You don't want to end up stuck out overnight without shelter when it's raining, even in summer... Lots of things can prevent you from getting to a hut (track damage, slips, flooded rivers, misjudging times/terrain, injury, ...). And aside from the sandflies (no mosquitoes at least), summer is a great time to camp.
Re Great Walks: I personally would avoid them in the summer, too much traffic, huts need booking, restrictions on camping etc. There are so many other trips which are just as good, but get a lot less traffic. IMO they're categorized and marketed as 'Great' Walks because of the high standard of the tracks and huts, and suitability for a wider range of walkers.