No, no... no nest boxes. The goal is to
not have them breed. I always say leave the breeding to the breeders. A good breeder has the experience to know how to care for their birds to ensure proper health of both the parents and young. This means meeting proper dietary needs, vet care, testing for disease, and also the proper handling and management of both the parents and chicks.
The destroying of the nest could be a few things - a lot of birds like to forage and chew, so you need to make sure they have enough to keep them occupied. I know there are many forums posts, as well as youtube videos, and websites that can give you an idea about good cage setups, care, and enrichment for the birds.
As for the plucking behaviours, I'm going to go back to my original thoughts (and the points others have made) about either needing a bigger cage, two separate cages, and enough entertainment to keep them happy. Stress in birds can present in a lot of different ways - aggression, over-preening, excessive screeching, forceful mating advancements, etc. If they don't have enough space to get away from each other when needed, or other things to keep them occupied enough, it can cause problems.
I don't think anyone can give you a guarantee about what will work... it's probably going to be a little bit of trial and error. You may have to put each pair into separate cages with enough space and things to occupy them, or you may need to separate the males and females (which means finding out what they are). You might be able to get away with one really large cage with enough space and enrichment. Worst case scenario I imagine would be having to separate the plucker (if it continues to go after any bird despite the right conditions being met), and put it in a cage right next to other others. (Someone feel free to correct me if this is bad advice). Did you say they were in a different cage before? Do you still have it and is is big enough for two birds? Or maybe you can get a second cage for cheap off kijiji (make sure to get the right bar spacing and properly sanitize it) to try things out?
And I know it is hard, especially having four of them, but it's going to be worth having them checked by an avian vet (especially mr puffy). They would probably be able to give you better suggestions on how to help them.
Best of luck, I really hope you can find a solution. They are beautiful little birds