Those are not eggs. Men do not have eggs, but women do. Your parts are called testicles or testes, and the bag of skin they are in is called the scrotum.

One testicle being lower than another is normal.

13 centimeters is 5.1 inches and that is just around average, if that is your erect size.

If the pain is severe or causes problems working/walking/sleeping, then go to a doctor.

That's perfectly normal to have ocassional pain. Try masturbating and being sure you have a good satisfying orgasm. If this all works, you are probably fine. If it does not or you seem to have excessive pain, ask your doctor, don't be shy they get this a lot.