do you sweat alot while you exersise? and do you wash regulary? not washing properly can cause your pours to become blocked and sore? hope things get better!
Doing pushups wouldn't effect ur armpits.. see to see ur Dr... That would alleviate all ur concerns and get to bottom of issue..
Don't know if you can get it from exercise. But it is usually a sign of swollen lymph nodes in that area. You should have it checked by your doctor. I have had these before and it led to nothing. But you are better to be safe then sorry.