Help deciding on tramps for mid November/December

Hi, we are a couple with a toddler who we will be 4 weeks in NZ traveling with a campervan and stopping as often as possible for some tramping =D We have some good hiking and trekking experience, but we are still adapting to hiking with a baby. If all goes as planned, we will have "practiced" a lot here in Norway this summer before we go to NZ. We will start in mid November, finish in mid December and we are now trying to draft a first itinerary specially to decide whether we start in Auckland or Christchurch, and which parks we try to include (so we start booking the internal flight and campervan). So far, we spotted some regions and hikes, see below, but we are unsure about the weather. - Tongariro national park, for day trips such as from Whakappa visitor center to Tama saddle and back; and Mangateppo road to Mt Ngaurhoe or Central Crater and back. - Abel Tasman: the coast trekk and Tableland Circuit: - Nelson Lakes: around St Arnaud - Arthurs Pass: Cass Lagoon Saddle, Avalanche Peak, Goat Pass - Mt Cook: Miller Hut - Mt Aspiring: Routeburn to Harris Saddle, Aspiring hut and day trips - Fjordland: are there some nice day hikes there? I mainly found info about great walks and long trekks. How bad is the weather in Arthurs Pass, Mt Cook, Fjordland and Mt Aspiring at this time? What can we expect? Heavy rain? Light rain? The whole day? Whole weeks? We dont mind getting some rain, but it is not as fun to pass the treeline and not see a thing =D How are the snow conditions likely to be in Mt Aspiring, Miller Hut, Arthurs Pass and Tongariro? Can some of the tramps of those "areas" be walked without "snow equipment" (crampons, ice axe, etc)? At last, would you recommend me to start in the North or in the South? Any region to leave for the latest moment (towards mid December)? Thanks in advance and looking forward for visiting NZ =D