Not working for Norwegian but have researched a little so will try to answer what came up:
1. What are the different Norwegian outfits at the moment (NAI, NAN, NAS) and callsigns?
NAS - Original Norwegian Air Shuttle AOC, in Norway, 787's currently on this AOC, NAX/DY, "Northshuttle"
NAN - Norwegian Air Norway - New AOC in Norway for future purposes (may replace NAS AOC), NAX/DY, ?"Northshuttle"?
NAI - Norwegian Air International, Irish AOC, Currently most 737's based there, IBK/D8, "Nortrans"
NUK - Norwegian Air UK, British AOC, apparently 787's will be moved here also.
NAA - Norwegian Air Argentine, pure speculation but looks like something is in progress.
2. Is there only one agency, e.g. OSM, providing pilots to Norwegian, unless you're core?
No. Initially you deal with Rishworth (Sweden). Rishworth have created a subsidiary in the UK called "Global Crew UK" which will offer you a 3 year contract and be your employer.
After three years you are "moved" to another company called OSM UK (affiliated with Norwegian Air) and get a new contract which is (maybe) unlimited. The OSM contract is better (65h overtime threshold, loss of license, parttime availabl according to company needs etc...).
3. Are the contracts local, in compliance with local labour laws and policies?
I can only speak for the 787. Yes, Global Crew UK is a "normal" UK employer in front of the law. You are employed and social security/taxes are applied.
4. If you're hired by an agency, what happens if you're sick short-term?
Rishworth/Global Crew pay for private health care insurance, sick pay for 1 month. Additionally there is an insurance in case of "loss of income". Do not know more details about that.
5. If hired by an agency, what happens if you are sick long-term. Any insurance in place on top of local welfare system?
See above.
6. If hired by an agency and loose your medical, any loss of license in place to give you a chance to restructure your life?
Not with Global Crew, I've heard OSM provide LOL.
7. If hired by an agency, any pension contribution from the company on top of local welfare system?
There is a minimum required in the UK (1%?). OSM contract features 5% (word of mouth info only).
8. If hired by an agency and you are to have children will they, without discussion, allow you the maternity/paternity leave that you're entitled to under the policies of the country in which you're based or live in?
Probably yes as it is part of employment law in the UK. Wouldn't "experiment" too much with labour laws when on a time limited contract.
Most people I've spoken to are happy on the 787, good bunch of people, fun at work. But they all also emphasize that the working conditions have to improve!