Hi Henriette,
great news on your husband securing a job.
I guess which visa you apply for depends on how long the company will wait, probably not that long so I'd guess a work visa followed by SMC if possible (but I seem to remember there's a salary threshold as well as other criteria). I guess depending on your ages you might have other choices - I know there are more knowledgeable people than me on here about visas so hopefully they'll chip in!
Like Lou76 I agree $55 K for a family of 4 doesn't sound fun, but perhaps others do it without any problems, we're all different
I'm guessing that's before tax? So you'll be seeing less. However, if it's what the market is offering for his salary at his level then you kind of have to start with that. I've been here a year and changed job once which has helped increase my salary so you could consider he's not going to be on the initial salary for ever if he's got job skills which are in demand. But that's a gamble of course.
The 5k relocation sounds good...though it won't go far.
Wellington is a great city and easy to get about where ever you decide to live.
Depending where you live will depend the schools you can consider. I don't have kids, but friends recently did a recce and part of that was checking out the schools - it's interesting how important that visit was to their decision making. I also remember them saying there are lots of primary schools.
Hopefully some parents will be along to answer soon
On your belongings I think it's tricky to say. Personally we brought everything, we still had to buy a few bits and pieces before the container arrived because we had to wait 3 months. Not sure how long you'd had to wait for yours but I guess you need to consider can you "get-by" for a few months with limited items or would it be better to fresh? Without savings that could be tricky and you'll find you have other "start-up costs" to take into account, like your bond on a rental for example.
Good luck in your decision making