@Maverick's answer already gives you the Biblical basis of a "yes" answer, but if you want to move from a bunch of textual statements (though indispensable) to processing further philosophically to envision how these truths are embodied in Jesus, this answer is for you.
The theory
There are 3 resources that can help you:
Fr. Thomas J. White covered this precise question about Christ's human knowledge and self-awareness in his Tough Christology Question interview starting at minute 25:28 ("the relationship between Christ's divine and human knowledge) followed by Christ's beatific vision and "self-awareness" starting at minute 31:47.
Fr. Dominic Legge lecturing on The Church Fathers and Aquinas on Christ's Human Knowledge.
Dr. Eleonore Stump explaining philosophically the relationship between Jesus's divine and human nature in a 12-minute video interview (transcript here).
Basically, Jesus is a 100% divine person possessing his divine nature completely, but most of the time chose not to access it (i.e. not operate in it), preferring to interact with the world through his human nature which is also 100% his since the human nature is added on to his divine nature upon Incarnation. So Jesus can make non-moral mistakes as a regular human being, as well as can be ignorant on certain areas such as not being able to speak English or to know nuclear physics. The analogy given by Fr. Thomas White is that of a person who IS able to play a violin concerto but CHOOSE not to perform or even to let other people know that he can play violin.
But of his mission as Savior of Humanity, this is too important to be left out from his awareness. While we do not know the extent of his knowledge (example: in his human nature Jesus might not know (or want to know) the date/time of the end of the world), Fr. Thomas and Fr. Dominic above argued that Jesus must have had Beatific Vision from babyhood signaled by John 1:14 ("full of grace and truth") which enabled him to receive all revelations that the Holy Spirit wanted to inspire him and to transmit via prophetic revelation to his human mind. Therefore, in areas where Jesus MUST know prophetically to fulfill his mission, Jesus (as a human servant) accessed his divine nature for the necessary knowledge and assurance of God's plan and will for him while on earth. Beyond knowledge about his mission, out of solidarity Jesus might have accessed only the knowledge that we need to know for our own salvation.
Answering your questions
The following is largely a speculation but helps to show that your questions can be answered with logical consistency and coherence within the bounds of orthodox Trinitarianism.
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"?
For Trinitarian Christians, it's very important to see Jesus as the pre-existing Word taking on flesh (John 1:1-18), so the human nature is added to the Divine person. Thus, it is not symmetrical: in His divine nature, He always knows about the two natures, but when operating in his human nature, his baby soul might need to grow until at least his teenage years for Jesus to fully realize that he is a divine person, relying on the prophetic revelation he receives as a human being (for example, when hearing the voice from heaven and seeing the symbol of the Holy Spirit during his baptism). Due to his immaculate conception untainted by original sin, he would not have the messed up psyche we have, which would have blocked direct intuition of the Father loving him and communicating to him since babyhood. But we don't know at which point of his life on earth he knew of his divine nature for sure (since the Bible didn't tell us explicitly), but by the time he started his ministry he must have known since it is critical to communicate his role as God for human salvation. Otherwise, saying that he is the lamb who takes away the sins of the world would be ludicrous / megalomaniac if he didn't know he is God (listen to how Fr. Thomas White explained it).
If so, why was it that when the fully man Jesus prayed, he didn't pray to his divine nature but prayed to "the father"?
In his human nature, doing as an example for us, in his prayer he must have been depending and worshiping the Father. Even though the Father is technically of the same essence as his own Person (the Word), remember that there are 2 minds in Jesus, so his human mind (which is in time) prays to his divine mind (which is outside time). It is a human to divine "I-Thou" relationship, just how we would relate to God the Father. Furthermore, Trinitarians postulate immanent relations among the 3 "persons" of the divine essence, so as Son incarnate, Jesus always assumes the character/role of the Son in relating to the person of the Father, a divine to divine relationship. The same Father-Son relationship was made visible in the flesh for our benefit.
Did Jesus (the man) take advantage of his "God side/nature" in order to have power to perform the miracles that he performed?
Of course. As man, he performs the miracle that as God giving him the 1) direction for the right occasion to do it, as well as 2) the power to do so. If Elijah could bring back the dead by the power of God, Jesus could of course do that too.
