Is it normal for breast (1)to become swollen & tender with armpit cyst? My doctors refuse to drain them but they recur extremely often if they go away.

Perhaps see derm. Can't be certain from your description, but it sounds like you might have a dermatology condition called hydradenitis suppurituva, which involves recurrent cystic lesions in the armpits or groin or buttocks area. Depending on the diagnosis, there may be some medications or other treatment options available.