It was suggested in another thread that one can choose to be a believer or not.
Is it a choice, is it something inherent to one's psyche, or is a product of one's environment and experience?
I think that there is a neglected need to better define and explain what would constitute belief (more specifically, god-belief). There are also repressed and closely related needs for better vocabulary and better acknowledgment of the role of emotional responses in the continued use of god-conceptions.
No matter how insistently many people want to treat "fear of god" as if it were one and the same thing as actual belief, the two stances are arguably of opposite natures and even mutually exclusive (as well as with atheism). At some point we will also have to recognize the urgent and dire need for gauging the mental health of believers, which varies a lot indeed and can't be very well be neglected in current times.
All the same, I will attempt to answer your questions now.
God-belief seems to be impossible for some people, perhaps to a very high percentage of all people even. But peer pressure can be so overwhelming that many people go through their whole lives without ever having much opportunity nor encouragement to truly consider whether they are believers.
As a matter of fact, whole communities (from nuclear families all the way up to theocracies) are often intentionally, deliberately ill-equipped to even seriously consider whether their people are clearly believers, disbelievers or something else. Very often the social and presumed religious institutions shape themselves in order to avoid answering or even asking those questions. I have personally met anedoctal evidence that Catholic Catechism apparently neglects to even mention that atheism exists. That may be typical; the point is very often to promote the appearance, not the belief proper.
So my tentative answer is that god-belief proper is probably rare, even very rare. But many, many people are taught from too early an age that we are expected to profess belief no matter the actual facts and feelings - if for no other reason, because the social environments lack the interest in learning to deal with other stances.
We may very well lack any choice in actually being atheists or instead theists. But many people are taught to respond to the matter in certain specific ways for the benefit of their peers and their own perspectives for acceptance. Resisting those teachings and the hypocrisy that far too often comes with them is something of a choice even if belief and disbelief proper probably are not.