Of course NH has plenty of custom home builders who can build outside the realm of capes and colonials. Your questions about stone and slate can be better answered by talking to local builders and contractors.
I do know that the price of building materials has not dropped the way the housing market has, and labor is not cheap around here, so if you are building a custom home from scratch be prepared to spend some serious coin. You might be able to find a decent deal on raw land, though.
Also, if you are not familiar with the region I would proceed with caution. These are just a few things to consider:
First off, obviously due to the winters you can't build year round. You'll most likely need to have septic/well installed, and the town will need to approve your septic plans. If you don't have electric lines on the street for hookup you'll have to pay to bring those in. If you hit ledge when they start digging to put in foundation (this happened to friends of mine) you have to blast it (was a HUGE unexpected expense). Be prepared to have the process be more complex and expensive than it seems on paper.
There are plenty of houses in NH that aren't capes or colonials. Lots of ranches, craftsmen, split levels, and even some moderns. And not all capes and colonials have traditional interior layouts.
However, the first thing that came to mind when I read this post was
'When in Rome......'
Just sayin'
