Talk to several therapists before you visit one. Wise people choose their friends carefully, use twice as much discretion when choosing any type of health care practitioner. Consider ending the session if you are still uncomfortable and can't relax- many therapists will not charge if you terminate your first visit because you're uncomfortable. They might even suggest a different therapist. Any session should be preceded with a health questionnaire, as certain conditions preclude massage.
Laws vary from state to state.
That depends on the techniques employed by your therapist. If you do get undressed, the therapist will leave the room while you undress to level of your comfort. You will then and get on the massage table, and cover with a sheet or towel. A good therapist will keep you properly draped throughout the session, uncovering only the area in which they are working. A State Licensed or Nationally Certified Massage Therapist works only on muscle tissue, and even then should only work on the muscle tissue that you are comfortable having worked.