I haven't seen any hills and i live in NZ
But yes you need 20inch gear if you want to take it easy
as for backpackers as mentioned above they all have cooking etc but the backpackers have ports and pans plates etc and the beds generally have duvets ( or are they call comforters in the US ) and you would problem fine it hard to go less then 50 km or worse case 100km without finding a backpackers so it depends on the costs that you want to incur sometimes backpackers are cheaper than a tent site if there is only one of you of course you can free camp if you have enough gear
It will be warm enough in January ( maybe 20 to 28 ) but it may still rain and its also school holidays some of the best places will be packed
And the other poster was right the south island is quieter and the scenery is a lot different in fact like a different country but in saying that it does depend where you are in the north island and what roads you take all the main ones are sealed in the north but there are ones that are not and some with highway numbers are not sealed so look at your maps carefully
If you need a place to stay or some more info let me know same goes for anyone else on here
cheers