Why do some female singers breathe out [and] in on the mic and what is
this called?
This is called "panting" and it is generally only done on purpose in pop-music. When it is done intentionally, it is usually to simulate the sounds of sex and the female orgasm. Sex sells and it is a prevalent theme in pop (before it became a musical genre "Rock & Roll" was jive-talk/jazz-slang for sex. Back when jazz was "pop.") Otherwise, rhythmic breathing is sometimes used as an atonal, percussive sound - which could possibly be argued as being termed "vocable" but, I think that's a big stretch of the term's actual meaning.
If panting into the microphone is unintentional, it is for the purpose of thermoregulation. The only phrases that apply to this are "lack of breath control" or "losing one's breath."
Are there any techniques for calming the breathing down for a female
pop singer/dancer after a concert? If you have any instructional video
of how you do it?
Techniques for catching one's breath are no different for a singer than for an athlete or any other human being on Earth, for the matter. There are techniques for increasing and enhancing breath control. A simple online search for 'breath control' will turn up lots of results.
What is it called when a female singer moves the mic away from her
mouth in concert to breathe out and in and then move it back? I have
seen some singers do it poorly. They breathe out on the mic, move it
away and then move it back to breathe in.
Again, this is simply called "breathing" or "catching a breath." There is no technique involved and doing so does not require any talent whatsoever... it is the most natural and involuntary response when one's lungs run out of air. Any singer (male or female) that moves away from a microphone and back in order to breathe is simply either attempting to keep the breathing noise out of the microphone or they are just adding a bit of 'drama' to the performance and/or emphasis to the lyrics...
If a singer is unable to keep their breath out of the microphone (unintentionally,) then it is likely due to fatigue. Otherwise, they probably don't care about the breath noise and by moving away from the mic and back they, again, are just adding a bit of 'drama' to the performance, adding emphasis to the lyrics, or simply feeling the music and expressing that fact with body movement.
