I gather that Maori is a uniform language NZ-wide. Was this the case historically, or have a number of distinct languages or dialects been unified in some (artificial?) way to produce an official indigenous language?
Yes, and no. Maori was basically one language, but with significant regional differences. Aoraki is an example of the most prominent of such. The Maori spoken in the South Island (te Wai Pounamu) uses k where North Island Maori uses ng. Growing up in Rotorua, a North Island city heavily dependent on Maori culture for its economic well-being, I learned the Maori name for Mt. Cook as "Aorangi." I have friends who grew up in the Hokianga, a region in the Far North of NZ, one of three areas in NZ where it is still not uncommon to meet people whose first language is Maori. They tell me that they occasionally have difficulty with the Maori spoken by those from the East Cape region, while Tuhoe, spoken in a geographically isolated region is probably the most distinctive variety of Maori. You are absolutely right in assuming that there has been a synthesised hybrid version created in the interests of standardisation. My friends tell me that this "artificial hybrid" can be almost unintelligible to their ears.

I was also surprised to hear that there is a Maori equivalent for every city or town in NZ. I understand that traditional places of significance would have an original Maori name.
Getting back to NZ, I am intrigued by the existence of a Maori word for cities and towns that have sprung up in the last 200 years. Are you now applying a traditional Maori word to refer to the region occupied by the "new" settlement, and is there really a one-to-one correspondence between them, or have Maori names been invented recently to cover the gaps?
Generally, what has been done is to apply the ancient Maori name for a region to the prominent town of the region, if that settlement is of European origin. I live in Hastings, which is called Heretaunga, after the plains on which it stands. Nearby Napier is called Ahuriri, a name refering to a specific locale close to its port, that has been extended to include the entire city. As I understand it, this is the sort of system that has been used nationwide. I don't think that there were any regions of NZ which had not at least been visited by Maori, and henced named by them, before the arrival of Europeans. A lot of towns in NZ already have Maori place names, and these towns have not had English equivalents invented for them.

One final note on your mention of the challenge of naming indigenous peoples. The following is a very thoughtful, concise, summary of the application of the term "Maori" for the indigenous peoples of Aotearoa.http://maorinews.com/writings/papers/other/pakeha.htm