Upvote:0
My question is, will there be any immigration checks on our route to and from Paris?
No. Both Norway and Paris (well, France) are part of the Schengen Area. Quoting wikipedia.org (emphasis mine),
The Schengen Area is the area comprising 26 European countries that have abolished passport and any other type of border control at their common borders, also referred to as internal borders. It functions as a single country for international travel purposes, with a common visa policy.
Upvote:5
Under the Schengen agreement, “systematic” border checks have been abolished. In practice, it's very unlikely that your passport will be checked by the police but it does not mean that there cannot be any check or that the airline will not ensure you have appropriate documentation (in particular, Ryanair is notorious for that).
Legally speaking, whether a border check is likely and whether you are in fact allowed to travel are distinct questions. Holding a valid type D visa or a residence permit is definitely enough to travel to other Schengen countries or leave and reenter the area. I don't know if the “approval letter” is formally enough but it's possible (I know that similar documents from other country do authorize travels in other Schengen countries).