Yes, the barred and/or cuckoo gene is passed from hen to male offspring only when mated with a solid male. All boys will hatch with a white dot on their head. :)
I don't think this is merely from feather picking or mating. There looks like there is a greater issue, something dermatological going on. Considering that you are not seeing any evidence of poultry lice or fowl mites, I would treat for depluming mites and maybe use a hen saddle while healing as...