1

Depends what type of accommodation but with a baby, you could prebook as you go to make sure they have a cot or whatever for you.

2

I would book for wherever you will be towards the end of your trip - especially if it's Kaikoura.
We were in Kaikoura around a long weekend; it didn't even occur to us but when we decided we'd like to stay an extra couple of days, we found most acc'm was heavily booked - especially anything that would suit us with a toddler, and wasn't super expensive.
Earlier in your trip you should be ok as you go.
Just a couple of tips - when you are booking, make sure they do have a portacot, and ask if they have a proper mattress for it, and bedding if you need it. We have found quite a few places will hire you a portacot, but all it has in it is the hard base that comes with the cot. I'd be pretty uncomfortable sleeping on something that hard and wouldn't do it to a baby.

ETA I did say a couple of tips. If you are booking in a holiday park or similar, it's worth having a look at where the outdoor lights are. After trying and failing to get a good night's sleep in a cabin with a motion-activated search light right outside going off at all hours of the night, we now pay close attention to where and what the night lights are around the place.