This season seems a bit freaky so far weather wise. I am in a central North Island ski town and from the sounds of it Melbourne is colder atm. We still have really nice days and the fire isn't even required most days. So I don't think that it would be that much colder in Wellington. Expect for the wind - when it blows in from the south it is really f***ing freezing!!
The central shopping city of Wellington is flat, it is only the suburbs that are hilly and there is a good public transport system so you wouldn't have to walk too far.

It would be very feasible to do a weekend trip to the South Island - you can get a morning ferry over and go towards Nelson via Queen Charlotte Bay for a night. You could probably get down to Kaikoura where they do whale watching tours too (beautiful spot, between southern alps and a rocky shore beach). If you didn't linger around too much you could make it as far as Christchurch, but the central city is still broken Or you could stay in North Island and go to Martinborough which is pretty and has quite a few wineries (possibly not that fun when preg though). I would suggest a day trip to Kapiti Island but it is a no-go with SPD. Or the Central Plateau! See some volcano's??

In Wellington you can go to Te Papa the national museum (bit of a let down tbh). There are also a lot of small production plays at Bats Theatre if you are into that sort of thing. If you are lucky you might see the guy that wears only a blanket sitting in Manner's mall.