I’m a little confused. Do you only have one hen? When a hen goes broody, she stops laying eggs. If she is spending all her time in the nest yet is still laying eggs, something just doesn’t sound right. Either she is not broody or another hen is laying those eggs in her nest.
A hen stores sperm in a special container near where the egg starts its internal journey through the hen’s internal egg making factory. Eggs are normally fertile for two weeks after a successful mating.
The inbreeding question gets tricky. Whether or not it is bad depends on what genetics are already in the chickens being inbred. If you have bad genes in the chickens those will be emphasized. If you have good genes those will be emphasized. What is bad to some people may be good to others. I like a variety of egg shell colors but if you are breeding to a standard, you want all the egg shell colors to conform to that standard.
But I think what you are worried about is that you will be breeding freaks, deformed chickens. Inbreeding lessens genetic diversity. You need a certain amount of genetic diversity for a healthy flock. Over the generations as you inbreed you concentrate the genes in the flock. There are several different techniques you can use to keep genetic diversity up, spiral breeding, pen breeding, or occasionally introduce a new chicken (usually a rooster) into your flock as a few examples. There are other techniques.
From a genetic diversity viewpoint, there is no difference in breeding a mother-son, father-daughter, or brother-sister. The reason you often see people pushing father-daughter or mother-son instead of brother-sister is that if you are trying to give certain traits to the offspring, you can control which of these traits are passed down better by breeding a superior chicken to their offspring. The traits you want will be better concentrated. But as far as making freaks, there is no difference genetically. It just depends on which genes are in the individual chickens involved.
How do you handle all this in a small backyard flock? Select which chickens you want to breed. If a chicken has traits you don’t want in your flock, don’t allow it to reproduce. How you do that is up to you. Just keep your best chickens as breeders and you can usually go four of five generations before you start to have genetic problems in the flock as a whole. You may get an occasional problem, but these are the ones that don’t breed. Then you just bring in a new rooster and you are good to go for another four to five generations. That is one method farmers have used for thousands of years to keep small flocks with only one rooster.
Welcome to the forum. I’m glad you found us and hope you stay active. But I’m still confused as to what is going on with that hen.