This is a very important question which demonstrates much thought and insight. It probably gets deeper into doctrine than you can initially imagine, but I will attempt to give you a basic answer.
The short answer to your question is: No, Jesus did not inherit the sin nature from Adam. Jesus came to succeed where Adam failed. Jesus is called the "last Adam" (1 Corinthians 15:45) and the "second man" (1 Corinthians 15:47). Jesus testified that He came "to seek and to save that which was lost" (Luke 19:10). What was lost in Adam was to be won in Jesus Christ. However, in order to do this, he only needed to start where Adam started, not where he ended.
Much of the purpose of the virgin birth was to allow Jesus to be fully human but without the Adamic nature. Joseph was "the husband of Mary, of whom was born Jesus, who is called Christ" (Matthew 1:16). Joseph was not the father of Jesus. Jesus did not receive the sinful nature of Joseph. However, He was born in the natural body and in the full likeness of sinful flesh. Romans 8:3 states, "For what the law could not do, in that it was weak through the flesh, God sending his own Son in the likeness of sinful flesh, and for sin, condemned sin in the flesh." He was born in the flesh, but the inner sin nature was not His nature. He was only born in the "likeness" of sinful flesh. If He had had a sin nature, John could not have said of Him, "in him is no sin" (1 John 3:5) and Paul could not have spoken of Him as the one "who knew no sin" (2 Corinthians 5:21).
This brings up the question of temptation. A common, though sometimes denied, teaching concerning Christ is that of His impeccability. This is the teaching that Jesus Christ could not have sinned. Thought the temptation was truly given, He could not have submitted to it. The motivation behind this teaching is the horror in even contemplating the idea of Jesus sinning. But this approach by itself does not answer all the problems concerning the temptation of Christ. The Bible teaches that Jesus was "in all points tempted like as we are, yet without sin" (Hebrews 4:15) and that because of what He suffered in temptation, "he is able to succour them that are tempted" (Hebrews 2:18). This points to a very real temptation. How could the temptation be real if He could not have submitted to it?
The solution lies in seeing two aspects of the temptation. Physically, Jesus could have sinned. There was nothing hindering Him submitting to the temptations of the devil. He had the same opportunity that Adam and Eve had in the Garden of Eden. From this aspect, the temptation was real and His victory over temptation was real. He truly suffered in temptation. However, there is another aspect. Morally, Jesus could not have sinned. In this manner, He was truly impeccable. His very nature would not allow Him to sin.
It is just as if someone showed you a million dollars that they had found and offered to give you half of it if you did not report it to anyone. Hopefully, you would say, "I just cannot do that." But by that statement, you would not mean that you physically could not take the money and run. Rather, you would mean that your moral makeup would not allow you to take it. However, while you were saying no and refusing to give in to the temptation, your mind could still understand the appeal of having half a million dollars and could think of many things you could do with the money. In other words, though your moral makeup would not allow you to take it, it would be a very real temptation. Though Jesus could not allow Himself to submit to the temptation to turn stone into bread at the suggestion of the devil, He could feel the need of the body for bread after forty days of fasting.
