Was Gautama Buddha a believer of the religious beliefs prevalent in his time and in his cultural context?

The Ariyapariyesana Sutta: The Noble Search is a partial description of his quest for enlightenment.

It contains passages like the following,

"Having thus gone forth in search of what might be skillful, seeking the unexcelled state of sublime peace, I went to Alara Kalama and, on arrival, said to him: 'Friend Kalama, I want to practice in this doctrine & discipline.'

"When this was said, he replied to me, 'You may stay here, my friend. This doctrine is such that a wise person can soon enter & dwell in his own teacher's knowledge, having realized it for himself through direct knowledge.'

"It was not long before I quickly learned the doctrine. As far as mere lip-reciting & repetition, I could speak the words of knowledge, the words of the elders, and I could affirm that I knew & saw — I, along with others.

"I thought: 'It isn't through mere conviction alone that Alara Kalama declares, "I have entered & dwell in this Dhamma, having realized it for myself through direct knowledge." Certainly he dwells knowing & seeing this Dhamma.' So I went to him and said, 'To what extent do you declare that you have entered & dwell in this Dhamma?' When this was said, he declared the dimension of nothingness.

"I thought: 'Not only does Alara Kalama have conviction, persistence, mindfulness, concentration, & discernment. I, too, have conviction, persistence, mindfulness, concentration, & discernment. What if I were to endeavor to realize for myself the Dhamma that Alara Kalama declares he has entered & dwells in, having realized it for himself through direct knowledge.' So it was not long before I quickly entered & dwelled in that Dhamma, having realized it for myself through direct knowledge. I went to him and said, 'Friend Kalama, is this the extent to which you have entered & dwell in this Dhamma, having realized it for yourself through direct knowledge?'

"'Yes, my friend...'

"'This, friend, is the extent to which I, too, have entered & dwell in this Dhamma, having realized it for myself through direct knowledge.'

"'It is a gain for us, my friend, a great gain for us, that we have such a companion in the holy life. So the Dhamma I declare I have entered & dwell in, having realized it for myself through direct knowledge, is the Dhamma you declare you have entered & dwell in, having realized it for yourself through direct knowledge. And the Dhamma you declare you have entered & dwell in, having realized it for yourself through direct knowledge, is the Dhamma I declare I have entered & dwell in, having realized it for myself through direct knowledge. The Dhamma I know is the Dhamma you know; the Dhamma you know is the Dhamma I know. As I am, so are you; as you are, so am I. Come friend, let us now lead this community together.'

"In this way did Alara Kalama, my teacher, place me, his pupil, on the same level with himself and pay me great honor. But the thought occurred to me, 'This Dhamma leads not to disenchantment, to dispassion, to cessation, to stilling, to direct knowledge, to Awakening, nor to Unbinding, but only to reappearance in the dimension of nothingness.' So, dissatisfied with that Dhamma, I left.

"In search of what might be skillful, seeking the unexcelled state of sublime peace, I went to Uddaka Ramaputta and, on arrival, etc.

In other words he studied with various teachers, and successfully learned what they were teaching, but wasn't satisfied (and therefore went to another teacher, etc.).

In other words, did Siddhartha Gautama arrive at his Enlightenment by BELIEVING in prevailing religious mythology and principles, or did he arrive at his Enlightenment by QUESTIONING, EXAMINING and REJECTING them?

I think he started by learning teachers' doctrines and practices, but (having understood them) finding them unsatisfactory.