Is it true more or less the "miracle pill?
Answers: If you're talking something like hormonal birth control, it can help some of those things, but the push button word is *can*. It's possible, but it doesn't work for everyone that way. There are also profusely of possible negative side effects to hormonal birth control, too. I would just refer to it as a "miracle pill", although there are some benefits for some population (other than the benefit of reduced chance of pregnancy).
Putting estrogen within your system does not get rid of excess testoterone. you should ask your doctor. Although, birth control can give support to acne and give you regular period. I don't think it help with mood swings.
it depends, yes, some birth control pills can do that, although probably not to the extent to which you are thinking. and yes, near is another that can disrupt your normal period. if you are interested, definately talk next to your doc about it.