I have read in multiple places that named characters don't really do assignments, however, I have also heard (not confirmed) that they can be assigned to scrap stations and still produce even when not there.
I don't have shops yet but i have one of the original 5 named characters from Concord farming (I think they mention she is a farmer and that is likely her default setting) and I have the old drug addict lady standing guard duty and she actually seems to do it.
As for your fluctuations in happiness and population ... have you noticed that the happiness goes down when the population goes up? or is it one goes up the other goes up? I am curious if there is enough "amenities" there to drive up the happiness, which causes the population to go up (I have noticed I seem to get settlers quicker the higher the happiness is) but that with additional settlers, there is not enough to maintain that happiness level so happiness drops and predicates a drop in population. If the rises and drops are not synced or are not in equal proportion then it could be just a lack of equilibrium in the game where over population causes unhappiness, unhappiness causes population decline which allows happiness to go up and attract more population ... rinse repeat. Maybe trying throwing down more seating area, spreading the sleeping quarters (I have heard that anything closer together than 1 bed per floor tile can cause unhappiness), add some lighting, radio, and television.
Not saying it will fix the issue but if you overkill on the amenities, maybe you can level your population off at a higher level? This is all just speculation but it seems to makes sense in my mind.