Did Jesus in his human nature, know that he had another nature (as "God the son") inside of him or that he had two "natures"?
Yes, if we mean simply as a human. Christ is one person; for Him not to know about the divine nature would mean that He didn't know himself because God has perfect knowledge of all things, including himself. Or to put it another way, to imply Christ knew one thing as as one nature but not as the other implies not a personal (hypostatic) union but a union of persons.
No, if we mean did Christ know his divinity via human knowledge because a human soul cannot have knowledge something above its nature.
why was it that when the fully man Jesus prayed, he didn't pray to his divine nature but prayed to "the father"?
Because prayer is directed to a Person not a Nature. Praying to his divine nature doesn't make sense (nor really would praying to Himself).
Did Jesus (the man) take advantage of his "God side/nature" in order to have power to perform the miracles that he performed?
"Take advantage" can be read in different ways. The power of miracles certainly did not come from his humanity.