You can have more then 3 flies, but can not fish more then 3 at once I beleive. You are correct in not being allowed to wear felt boots while fishing. I believe you are allowed to bring in flies, but they have to be clean and dry, they also must be declared (along with all fishing gear) on arrival. Bit more information here
This website here http://www.nzfishing.com/ gives a good overview of freshwater fishing in New Zealand. I beleive that the Salmon should still be running in March, the nights have started to cool and the trout should be slowing moving upstream for spawning. Its a great time of year for fishing.