Hi, I'm here. The best information is that which you have already had - the mfd site. Food labelling is awesome, food availability in the South Island is another matter. My advice is to stock up in the large centres at the "posh" supermarkets, you need the one's with organic ranges as a lot of the dairy free stuff tends to be organic to and that's why they stock it IYSWIM.
It really depends on what sort of stuff you eat, meat and veggies are plentiful and all NZ motels have kitchens so you can make your own safe food. There are several breads that don't have milk solids in them and lots of soya milk, but it is all yuk compared to the UK and you will die at the relative price! Remember we are a dairy nation. We eat millions of litres of ice cream annually and have many many cows, the tendency is to slip a bit of milk into everything. When I was milk free for ds there was 1 kind of biscuit (and only one brand) which was milk free - I hope she likes gingernuts!
As for eating out - well ds is not anaphylactic, or at least if he isn't we haven't discovered it yet, and we don't eat out anyway. We have one cafe that we eat at every now and then which has a vegan selection of cakes and will make a dairy free sandwich. Cross contamination would be an issue though unless you call ahead.
Also be aware that no comercial margarine is dairy free which means no sandwiches from shops - but they all put cheese in anyway so not an issue.
Anyhow, I have rambled on more than enough, but please feel free to ask specific questions or CAT me.