Jesus was "tempted in all ways as we are".the human "nature" is prone to sin the moment we know "right from wrong". this is in our dna, which is the direct result of the fall of the human race in Adam.
Jesus Christ was "TESTED", as the Greek world also means, but, as the God-Man, could not have fallen, as this would be impossible. Jesus Christ is two "Natures", the Divine as eternally God, and the Human, at His Incarnation, though this is sinless. He is One Person, not two. When He was "tested" by the devil, it was the One Person Who was, and not the "human nature", only, as God-Man, it is impossible for Jesus to have ever fallen/failed. To "test" someone is with the expectation that they MIGHT fail, which is not a sin. It is only a sin when we do fail.
I do not think that it is good to get into the philosophical here because the point of that passage is to demonstrate that Jesus went through temptations "as common to man," and can therefore identify with us when we ate tempted.
I disagree that Jesus had two natures. I believe that the point of the Incarnation, the method of redemption, and the hope of our resurrection us dependent on Jesus having one nature (a solidarity with man....the Word be coming flesh). I am not sure, however, that this is a disagreement (reading your post you kinda hit on my point - Spirit-flesh, God-man).
But Jesus did nothing of His own accord. He relied on the Spirit, He submitted His will to the will of the Father. We are to be that way.
Was it impossible for Jesus to sin? Yes. Just as it is impossible for natural man to please God. It is not a matter of ability (Jesus had the power to come down from the cross) but a matter of the will.....of desiring the things of God over the things of the flesh.