1) Speaking for normal situations
No, not any sort of pain can vanish when our minds are focused on other things.
In case of minor pain it is possible to pay attention to other things, get absorbed in them and then not feel the pain, because our attention is somewhere else. In these cases the wanting to experience the other sense impressions is greater than the pain.
But then there are cases with severe pain. When I say severe I'm talking about pain that is immobilising, crippling, so great that one can't sleep. In such cases the mind can't help but pay attention to the body, the body simply demands it and will override any other desires.
Simply put: sometimes the mind initiates what will happen next (is the cause), sometimes it's the body.
It depends not only on what the mind finds useful, important, valuable. But other factors, like the heaviness/greatness of the sense impression, are also relevant in 'choosing' what to pay attention to.
And sometimes, it's the body that decides, so to speak.
To give another example: urinating.
When the urge to urinate is small/minor, one can put this simply aside, ignore it; go dancing, have a talk, watch tv, whatever. But sooner or later the urge will be so great that it no longer can be ignored. The body will force you to go to the toilet, whether you want to or not. When in sleep you will wake up.
Same with the pain. If it's minor the mind will pay attention to something else if it wants to. But if the pain becomes too severe, so severe that normal bodily functions can't be performed, then the body will override any other desires/wants the mind could have and attention will be drawn to the bodily sensations.
So, no. One can't remove any pain by simply focus on something else.
This back-and-forth between mind and body as 'leader of the action' is one of the early stages of insight (the 2n of all 16) we gain when we practice meditation.
2) In 'not normal' situations
I'm referring here to deeper stages of concentrations.
When totally absorbed all pains vanish because the experience of the body has ceased completely. And with that the experience of physical pain will cease as well. This is the fact in the fourth (meditative) jhana. You won't feel breath anymore. Even the experience of the heart beat can be gone.
But: a) those stages are always temporarily; and b) you can't get into those stages with pain as severe as I illustrated above simply because the pain is so demanding and changing that the mind will never get still enough. (I'm a chronic pain patient. So, I do speak a bit from experience here. I also practice jhana next to vipassana.)
When it comes to those deeper stages of concentrations (jhana). You say you never been in them. I sincerely doubt it. In fact, in your answer to Sankha you explain a jhana perfectly: you didn't feel the pain until you were done dancing. As a ballerina your focus on your movements has to be absolute. You can't afford to be somewhere else with your mind or you'll loose balance. A voila: you have been in jhana when performing.
It's just not a (higher) meditative jhana, it was an 'ordinary one'. (Jhana are often misunderstood, especially by those not practising them.)
What I want to say is that you know what it is and there is no reason to think that you can't go there in meditation as well. In fact, I think that your habituation will incline your mind more towards concentration practice (samatha) than wisdom practice (vipassana). Something to look out for when you meditate, because it can present you with certain challenges.
Anyway. Hope this clarifies things a bit better.