I'm not so sure we have any active Kiwi boarders at the moment. I've never been to New Zealand, but I live right across the ocean from it and I have the power of Wikipedia at my disposal so let's see.
1. Is this a good idea?
Why not? They speak a rudimentary form of English and they're very lovely people. The culture is interesting, the scenery amazing and it doesn't seem like an awful place at all.
2. I was thinking the north island for no real reason. A wise choice?
Supposedly you want to be near either Auckland or Wellington (both on the North island) because elsewhere there is not much really.
3. If I'm there from January to June, what will the weather be like? I'm confused by the alternate seasons and have heard conflicting reports! I like warm weather more than cold weather...
The seasons are backwards in the Southern Hemisphere: Where the US and UK have Winter, we have Summer, and vice versa. if you come in January you'll see most of summer (gets up to 30 degrees celsius or 86 fahrenheit) and in June you'll see the start of Winter (gets down to 10 celsius or 50 fahrenheit and snows in mountain areas, but not in the cities as far as I know).
4. Should I study in NZ or Budapest (another option open to me)?
I'd go NZ. I'm actually considering going there myself sometime this year, we shall see. Budapest? What is there? It doesn't sound particularly intriguing. Wiki lists its most interesting points as a subway, a cave system, and a building. Hmm.
Also re naming:
The name New Zealand originated with Dutch cartographers, who called the islands Nova Zeelandia, after the Dutch province of Zeeland. No one is certain exactly who first coined the term, but it first appeared in 1645 and may have been the choice of cartographer Johan Blaeu. British explorer James Cook subsequently anglicised the name to New Zealand. There is no connection to the Danish island Zealand.