Upvote:3
Will be able to travel to the other Schengen states visa-free
No. Well, yes, but no. I mean... in practice, many / most of the time when you cross a Schengen border with car/bus/foot nothing much happens, sometimes not even a flight incurs a passport check. But anything goes. You cross the border, five minutes later the police is signalling the car demanding paperwork. I had a Budapest-Milan flight which ended in a passport check much to my surprise (a few days later a Rome-Marseille one didn't nor did a Basel-Budapest one a few weeks later). This Christmas driving from Hungary into Austria there was a passport check on the border but driving from Austria into Hungary noone gave a damn. So: to stay within legal limits and avoid huge problems, you need a visa.
or if not, do they normally accept an NZ RTD to apply for a visit (tourist) visa?
Absolutely. The 1951 convention document equals a passport in these cases.