No doubt a huge mess.
Here at TMDb we have more skilled moderators than myself to solve this issue.
Exactly why, as a user, I would solve it this way:

Now what with be the proper value for: Origin Country, Production Countries, Original Name,

Origin Country: Norway
Production Countries: Norway, Denmark (Greenland)
Original Name or Title: The used at the premiere in Sweden. The Icelandic title.

To complicate things further, the title is different in all of these regions.
In Norway the choose an English title,
in Sweden they use a Icelandic title,
while in Denmark it uses a title of an artifact in the series, which is not a dictionary word, thus not a language at all.

Norway Title: English title
Sweden Title: Icelandic title
Denmark Title: SDRUVS (an artifact which is not a dictionary word)

Also which of these values would change if the series were to debut in one of the other countries,

If the premiere took place in one of the other countries?
It would change the Original Title to the title used in the World Premiere country.

or if it aired at the same time in these regions?

If the premiere was simultaneous in all 3 countries at the exact same time, then I would use as original title the one used in the simultaneous premiere in the country of origin.
In this case Norway and not the one used in Sweden.

Now I will create one more problem.
What if the premiere was simultaneous in Sweden and Denmark, but not in Norway?

I believe that in this case the decision would be arbitrary.
And I would choose the title from Denmark. the SDRUVS.

In time:
SDRUVS - Tool used to macerate wild mushrooms with nuts and some reindeer fat, creating a paste used as a spice for ember-roasted herring.

Especially where the origination country uses a language which is not official in the country. (Like a German show using an English title.)

The language of the title is not relevant. For an example of a German language TV show using an English word/expression as the original title, see 4 Blocks.

A Norwegian production creates a series set on Iceland, where they speak Icelandic, however the series is recorded in Greenland (Denmark), cooperating with a local cast and crew.

Production Countries: Norway

There might be more co-producing countries, but you do not give us enough data. While the TV series if filmed in Denmark, it doesn't mean it should be added as a production country. For example, many American TV series are filmed in Vancouver, Canada (hiring Canadian crew and actors) because it's cheaper and the BC is beautiful. But they are very, very rarely American-Canadian co-productions. So does you made up series have any DK or other foreign production companies? Or at least was produced in association with a DK/international TV network?

a series set on Iceland, where they speak Icelandic

Spoken Languages: TBD

So this Norwegian TV production filmed in Denmark is set in Iceland. Which language(s) are the characters speaking in the original (not dubbed) version? Only Icelandic? Norwegian? Icelandic and Norwegian? In the TV section, the spoken language field is only used for the main (one) spoken language. But if two languages are consistently spoken throughout the series, you can add two.

To complicate things further, the title is different in all of these regions. In Norway the choose an English title, in Sweden they use a Icelandic title, while in Denmrk it uses a title of an artifact in the series, which is not a dictionary word, thus not a language at all.

If it's a Norwegian production, the Norwegian title (i.e. title used in Norway) is almost always the original title. One big exception is TV series produced in a foreign language for an international market, but it's almost always an English language TV series produced by an European country. I'm taking a guess that it's not the case here, but we're missing some data.

Original TV Show Language: Norwegian
Original Name: Norwegian title (title in English)
Norwegian: field locked blank
Sweden: Swedish title (title in Icelandic)
Denmark: Danish title (artifact title)

The series debuts in Sweden, before it airs in Norway and Denmark.

This is all covered by our TV guidelines. There are two choices:

1. Use the very first air dates in a foreign country/language. So, in this case, we would use the Swedish air dates, add Sweden as the origin country and the Swedish network to the network field.
2. Ignore the Swedish premiere since it's a Norwegian series. Instead use the Norwegian air dates, Norway as the origin country and the Norwegian network to the network field.

We typically use the second option:

Origin Country: Norway
Network: NO Network
First Air Dates: Air dates on the NO network

Also which of these values would change if the series were to debut in one of the other countries, or if it aired at the same time in these regions?

The only fields that might change are the First Air Dates, Original Network, and Origin Country. But, again, if it is 100% a Norwegian series, Norway should ideally be the only country of origin/network. We do not yet support international broadcast.

If the premiere took place in one of the other countries? It would change the Original Title to the title used in the World Premiere country.

No, by definition the Original Title is "usually the title of the original version when it is first officially released locally". Many TV series and films premiere in a foreign country/language, but it doesn't have any incidence on their original title.

OK thanks for taking your time to clarify. Based on your definitions I think the labels are a bit confusing. It seems to be that they should be "title in country" rather than "title in language". But I do understand the many aspects, like a country that has several languages, Canada has French and English, or a language could be official in several countries. Like English , French, Spanish, etc.

I'm not quite sure how the latter is used. As I can see that there are values for «Australian English» and «Canadian English» in the preferred language, but I've never seen a page where there is more translations than one English. In fact, I've never seen other than EN-US for the shows I looked up. While the dropdown for original language and spoken language have just English. I think it makes sense, I'm just trying to understand the definitions.

Depending on the actual definitions I think something similar should be done with Norwegian, which currently are separated into Bokmål and Nynorsk, as well as "Norwegian".