Regeneration is a biotechnology, not a natural process. It is granted to Gallifreyans when they become Time Lords (which involves hereditary rights as well as tests, among other requirements; Time Lords are the technological elite of Gallifreyan society).
Regeneration is limited to 12 times per Time Lord not because it is impossible beyond that, but a) because the "photocopy effect" gets worse each time and beyond 12 it's less reliably stable and b) Rassilon was a jerk, but he recognized that true immortality isn't good for anybody.
Now, closing in on your question specifically: regeneration can happen under three known conditions.
First, the High Council can force it.
Second, the Time Lord can voluntarily trigger it.
Third, it occurs automatically when a Time Lord is lethally wounded (though sudden death can interfere with or prevent this).
Regeneration is a process by which all the cells in the Time Lord's body are rejuvenated at once. It is somewhat like becoming younger, but the process also scrambles the Time Lord's DNA so that he changes appearance and his personality traits are reshuffled (not changed, just re-prioritized).
When the process is triggered on purpose, by the Time Lord or the High Council, the result of the regeneration is controllable.
Indeed, we've seen one Time Lord who seemed to be able to "try out" different bodies during the process before she settled on one she liked--although that could have been time-projections similar to the Valeyard.
But when the process is forced by mortal injury, it is uncontrolled and unpredictable. At best, the Time Lords' unconscious thoughts and desires at that moment may influence the re-prioritization of his personality.
So: generally, it seems that the Doctor doesn't have control over his appearance (except maybe subconsciously), because he only ever regenerates under stress.
If he ever chose to regenerate, he'd be able to choose his new image.
(The Second Doctor was forcibly regenerated into the Third by the High Council, and he was given a choice of new bodies; he rejected them all, so they chose for him.)