I'm going to be a bit pedantic, but wrt to this part:

  > because they speak an unique old Norse tongue

  They don't speak Old Norse

Note that "old" in the original quote isn't capitalized. They aren't saying they speak Old Norse, but a unique (and somewhat older) Norse tongue.

> we have other North Germanic languages (Icelandic, Faroese)

I could well be wrong here (please correct me if I am).

Wasn't Old Norse divided into Eastern and Western? With Icelandic and Faroese related to Western Old Norse whilst Swedish descends from Eastern. This would imply that Älvdalska would be related to Eastern Old Norse - hence not totally unique but still of great interest.

Not really, no.

Nowadays we have nation states and boundaries. We have a buraucracy, publishers and broadcasters located in the capital that influence the language spoken in that country. 500, 1000, 1500 years ago it wasn't quite like that. Things were more fluid.

Älvalska descends from whatever was spoken there, which was presumably closer to what was spoken in Uppsala at the time than to whatever was spoken in Hamar (in Norway some way west of Älvdalen). But there wasn't a ban on travel to Hamar, nor a state that lent prestige to the dialect spoken in the capital.

Yes it is. Old Gutnish is usually seen as an additional dialect of Old Norse.

I've only had a few courses about Old Norse, that were mainly about Old West Norse, so I'm not really an expert in this topic. I think we pretty much know the differences between Old Western Norse and Old Eastern Norse through the texts of that languages that were preserved.

I know nothing about the topic (besides English of course which is my native, I have only studied Spanish and Japanese), but this reminded me of a graphic I've seen: http://mentalfloss.com/article/59665/feast-your-eyes-beautif...