Itzjbean has summed-up everything you need to know and act on in her post, and you'd be very wise to take this seriously and listen to what we are telling you, as both she and I have bred parrots for many years, and every bird breeder is going to run into the situation that you're in, which is that you have put together a male and a female bird that simply do not like each other, and that probably not EVER going to breed together.
Birds are not at all like dogs, cats, rodents, reptiles, or any other type of "pet" where if you want to breed them all you have to do is put any male and female together and it will happen. Birds are very much like humans in the way that they decide who their partner/mate is, and who they are going to breed with. With birds, long, long before they ever even think about breeding, they must first find a bird of the opposite sex who they like and who they actually WANT TO bond closely with. Once they find a bird of the opposite sex that they like and that they start bonding with, it will still be a long, long time before they are bonded closely enough that they start mating, and eventually breeding. So if you think about how we as people meet someone, decide whether or not we are at all interested in dating them, then decide to start dating them, date them for a long time, eventually move in with them, eventually get married, and THEN decide to have children, that's very much how birds work when it comes to breeding them.
****The two birds that you have don't like each other at all. Right now I would classify them as two birds that don't like each other at all, but that are simply "tolerating" each other because they have no choice, they're trapped together and have to live with one another, and they are simply putting-up with each other, and have bouts of aggression and violence..The only outcome of putting two birds together that is worse than what you have with your two birds is where the two birds absolutely hate each other and are constantly aggressive and violent with each other, to the point that they are hurting one another and will eventually kill each other. It sounds to me like your birds have the potential of getting to that type of situation, since the female is already being aggressive with the male...that usually will keep progressing until one day she really seriously injures him...
***Forgetting that situation for the moment...You're birds are not going to breed, as they just don't have the type of bond with one another, and they are never going to get to the place where they are bonded closely with one another and where they want to start mating. And putting a nest-box in with them is not going to entice them into mating with one another, it's pointless to have the nest-box in with them, and actually is a really bad idea in your situation because not only are they not ever going to start mating, but things like nest-boxes, nesting-materials, etc. can still trigger hormonal behavior in your female bird, but in your situation instead of your female becoming hormonal and wanting to breed with her "partner" or "mate", since they aren't partners or mates and she is already being aggressive/violent with her, she could actually become more aggressive with him and actually end-up hurting him badly...or worse. So the very first thing that you need to do is to remove the nest-box from the shed, because again, it's definitely not going to get them to start mating, but what it might do is cause your female to really hurt your male.
The thing that you need to realize is that you could have just as easily brought home a second bird of the opposite sex, just like you did, but let's say that you just by chance chose a different bird than the one that you brought home...and in this scenario they may have bonded closely with one another. That's just how it goes. But what you need to understand is that there is absolutely no way at all that you can FORCE or ENTICE your birds to start mating/breeding. Their relationship with one another isn't anywhere near what it needs to be for them to breed...
*****And in the situation that your two birds are in, the best thing that you can do for the welfare and health of both of your birds is to completely separate them ASAP. For future reference, breeders who are attempting to "pair" birds of the opposite sex for potential breeding don't just put two brand new birds who have never seen each other inside of the same cage with each other! Nor should ANYONE, whether you are hoping to put two birds together to breed, or you simply want two birds as pets, you should NEVER, EVER put two birds inside of the same cage together the minute you bring them home, or the minute you bring home a second bird and already have a current bird. It's dangerous, unhealthy for both birds, both physically and psychologically, and to be very forward with you, it's irresponsible. The proper way to bring home a second bird, whether they are of the same sex or the opposite sex, and whether you want to try to breed them or they are just going to be pets, doesn't matter, you should ALWAYS do a 30-day quarantine of the new bird in a totally separate room from your current bird, and then once the quarantine is done, you should put their individual cages in the same room so that they can see each other and talk to each other, and so you can see whether or not they are getting along, or if they are just ignoring each other and don't care, or if they hate each other and are showing aggression. And then you need to make your decisions about housing them together in the future based on how they are interacting with each other while in separate cages but in the same room. You can't just put two birds together in the same cage and then just "see what happens"...
***All of that being said, the bottom-line in your situation is that your birds are most-likely not ever going to start mating, as they don't have that type of bond with one another, but that is the least of your issues. Breeding your two birds shouldn't be anywhere in your plan for them right now anyway, as based on what you are describing with their diet and the male's feathers, they are both most-likely suffering from some malnutrition and nutritional deficiencies, as well as other physical health conditions, such as possible liver disease, kidney disease, etc., due to their diet, as well as their level of stress due to their living arrangement and environment.
Your male's feathers falling-out the way that you're describing them is not likely a "molt" at all, but rather a combination of nutritional deficiencies due to an all-seed staple (very high in fat) along with nothing else but fresh fruit (very high in sugar which is turned into fat), along with high psychological stress due to being completely dominated by the female 24/7, and then also some degree of physical abuse to him by the female. I'm sure that you are aware that mental stress is extremely hard on birds, they don't handle stress well at all, sometimes to the point where they literally die due to nothing else but high, psychological stress. What you are describing with your male being not only dominated by the female in every way, but also being physically abused by the female, combined with poor nutrition is causing your male some potentially serious physical health issues. And your female is also going to be suffering from the exact same nutritional deficiencies and possible illnesses/diseases due to a high-fat, high-sugar, low protein, inadequate vitamin/mineral diet. So she is absolutely in no condition at all to start laying eggs, which is extremely hard on their bodies under the best of circumstances...Does she have access to both a Cuttlebone and a Mineral Block at all times, so that she's getting adequate Calcium? Even though your birds are not going to breed, it is still very possible for your female to start laying infertile eggs at any time, and if she is suffering from any type of nutritional deficiencies and she does not have access to ample Calcium at all times, she is at great risk for Egg-Binding, which is 100% fatal without immediate medical intervention.
What I'm trying to say here is that forgetting that your birds are not a breeding-pair and are most-likely never going to breed, that should not be what you are worrying about right now. Right now you need to take the nest-box out, separate your two birds so that the male is not constantly under so much stress and physical abuse from the female, and then you need to get both of them to a Certified Avian Vet or an Avian Specialist so that they can each have a wellness-exam, and so that they can each have routine blood-work done to determine whether or not they are suffering from any infections, any nutritional deficiencies, and so that you can see their liver function and kidney function. And you also need to start changing their diet immediately. They should be eating a fortified parrot pellet as the main staple of their daily diet, with a healthy, varied seed-mix that contains NO sunflower seeds, peanuts, or any other nuts in a very small quantity as a supplement to the pellets daily. And then they need to be given fresh veggies and dark, leafy greens every single day as well, instead of fruit every day. Fruit should only be given a couple of times a week and in small quantities, as it's loaded with sugar which is turned into fat. Fresh fruit does not at all replace fresh veggies...You need to work on getting both of your birds physically healthy, their diet nutritious, and both of their environments safe, comfortable for them both, and free of constant stress, and the only way to do that is to separate them each into their own cages with their own food, water, toys, etc. If you have to keep them in a shed instead of in cages, then you need to separate the shed into two separate living areas where they both have their own food, water, toys, foraging activities, etc.
