Not necessarily. It is not unusual to be using first names and be on formal term at the same time. Common at workplace. In English, once you switch to first name you are automatically iinformal. In Czech you can still be formal. Your boss would call you John but use formal VY at the same time. You might call him Mr. Smith.
Both. When used by a woman or about a woman as "můj přítel" or "její přítel" it almost always means a boyfriend unless the context calls for otherwise. For men it is mostly a very good friend, a trusted friend. I am not sure about the current language situation in gay circles though, but I heard that "partner" was more common for a "boyfriend" to be explicit.