I don't know that it demonstrates wisdom or folly on my part to jump into this conversation at this point...but here goes!
...however, is seems like Calvinism teaches that a man has no ability to choose God while he is lost.
I would agree. This is what Calvinism does teach. More on this in the next response.
Yet, after he is saved, he can chose to either obey God and live spiritually or he can chose, at times, to follow his old sin nature. Is this correct?
This is, again, what Calvinism teaches. This idea stems from Augustine where he outlines 4 options of theological/soteriological anthropology:
1. Man's state before the fall: Able to sin; Able not to sin.
2. Man's state after the fall: Unable not to sin.
3. Man's state after coming to Christ: Able to sin; Able not to sin.
4. Man's state in the eternal kingdom: Not able to sin
Augustine would argue, and I'd agree, that the only true free will (which, again, is not a libertarian free will) is exhibited in those who are in Christ.
The reason for this is the will is bound by the nature--a heart that desires "only evil continually"--before one comes to Christ. There is freedom to do only that which is in concert with the will. Since the heart is totally corrupted, the will is not "free" to go against the corruption--hence the Calvinistic understanding of why regeneration must necessarily precede our coming to Christ.
If it is, another question is, when a Christian does sin, which we all do, is this sinning doing what God wants him to do or is he acting outside of what God's will for him would be?
The answer here is yes and no. In one sense, it is absolutely true to say that our sin goes against God's will. This is plain in the text of Scripture (that which is not done from faith is sin, etc.).
However, we must also recognize the second aspect of God's will--that He ordains our sin to suit His purposes. So, might a Christian's sin--absolutely sinful and wrong and contrary to God's will as it might be and actually is--be "ordained" for a greater purpose? Sure.
Let me give an example: Let's say a Christian man has an affair. Is that sin? Absolutely and undoubtedly it is a horrible and wretched sin. Let's say that by God's grace (and his wife's) that the relationship is restored and blossoms and is strong and proper for the rest of his life.
Might this man be able to detect the signs of a coming affair in the lives of other men? Most likely. Will he be in a unique position to offer council and help to prevent other men from falling into the same sin as he? Absolutely. Would it be God's will to "use" that sin for the greater benefit of many others? Sure.
So there are two things working here--God ordaining the free and sometimes sinful actions of human beings to serve His purposes and display His glory.
So, the answer is not either/or but both/and. But, as Paul says in Romans, this does not give us either the license or freedom to sin that "God's grace may abound."
Blessings,
The Archangel