1. Your seniority is not so much based on what your staff number is, but the date you officially join the company, i.e. your first day of ground school, and it is this date that guarantees your seniority. This applies to all aspects of the company irrespective of fleet or AOC, so yes your assumption is correct.
2. There are no different branches apart from short haul / long haul. The only 'branches' are the different AOC's:
NAS/NAN -> B737 in Norway and Arlanda, B787.
NAI -> B738 & Max for all European bases, CPH and B737 US & French Caribbean bases.
NUK -> B737 & B787 based LGW.
NAA -> Argentina, which you can only apply for if you have an Argentinian passport.
3. If you are employed in OSL your employer will be PSN. As such, you will only be flying LN- registered aircraft. Likewise, if you're based in ARN, you will fly LN- aircraft, the exception is CPH which is part of the NAI AOC.
4. Once you join you will be given a training base followed by a release base, which you keep until the next base bidding. If you then bid for your preferred base and get it as your permanent base you cannot be kicked out from that base in favour of a more senior pilot. If however the base has a cut in production and people need to be moved, it's a case of LIFO, strictly according to the MSL.
5. The winter layoffs happened in 2014 where the last 60 on the MSL were placed on unpaid leave irrespective of base, seniority being the determining factor. Presently it doesn't look like this will happen again but if it were to happen again, then the MSL applies as above in point 4).
6. The old K-area contract as it was called then, which people refer to as core is not the holy grail anymore that it once was. All pilots are now agency employees, in the Scandinavian bases through Pilot Services Norway/Sweden/Denmark, and elsewhere through OSM. So no, 'core' doesn't exist anymore as such, but there has been no change in T&C's, meaning that if you're employed in Scandinavia, it will be on 'core' T&C's, but not employed by Norwegian but a PSx company.
I hope this clears up some of the questions.