The sand flies are bad in the fall too. I was at Abel Tasman and Punakaiki and getting bit. Punakaiki was worse as I watched the sunset and two of the little devils even left scars (full disclosure, I scar easier than the average person).
Adding to the Tongariro Crossing information - I had clients do it with a guide in April. It was a sunny day during the week and they were surprised how many people were on it. They were also surprised that it was much harder than they were expecting. It was a family of 4, boys ages 13 and 15 and are very, very active. They said it definitely should only be done by people who are skilled. That being said, there are tons of hikes in the area and you will find plenty you can do with the family.
Te Papa is an amazing museum and they have a big play area for kids, too.