Yes, you may file an EEOC charge if you are not hired because of your gender. You must do so within 180 days of the decision to deny you employment.
You can, but it is a tough case to prove. What prof do you have? Take your information to a local employment lawyer.
This answer does not constitute legal advice nor form an attorney client relationship. I am not your lawyer. If you have a legal issue in Maryland, Virginia, or the District of Columbia, you may contact me for a free consultation.