Contradictions in Genesis One and Two

Genesis 1:25-27 "And God made the beast of the Earth after his kind, and the cattle after their kind, and everything that creepeth upon the earth after his kind: and God saw that it was Good. and God said, 'let us make man in our image, and after our likeness, and let them have dominion over the fist of the sea, and over the fowl of the air, and over the cattle, and over all the earth, and over every creeping thing that creepeth on the earth. So God created man in his own image..."
Genesis 2:18 "And the Lord God said, "'It is not good that the man should be alone: I will make him a help meet for him'. And out of the ground the Lord God formed every beast of the field, and every fowl of the air, and brought them unto Adam to see what he would call them: and whatsoever Adam called every living creature, that was the name thereof."
  So were the animal made first, or man? And if they are two different accounts as indicated by the writing style and different ways of identifying God, and if they contradict each other, does that make them disprove the authenticity of the Bible?
Did you know that if you type in Explanations of Contradictions in the Bible from a Biblical Perspective you only get a few sites where it does exactly that out of the hundreds of hits?
It is noticeable that in the priestly source, it is Aaron's staff that produces water from rocks, splits the sea, and casts plague over Egypt, whereas it is that of Moses in JE.
  P sure is worried about the authority of the clerics. If I recall correctly, he was also the one to push God as a transcendent figure, inaccessible to the common people, who never hesitates to punish wrongdoers. Not exactly the kindest, most caring God.  
  This method of pushing God away seems contrary to what I recall of even the most basic traditions of other ancient religions, who offered a plethora of physical idols and cosmological events as proof of their Gods' existence. Is the Priest's account of such a vengeful god a way to make him more active in the Jewish people's life, since he was never seen in the presence of an object? (Not counting the Ark).