I agree that without works faith is dead, but that faith is the foundation. I will go over some of what I have come to believe. Most people believe something, about politics, where did I come from, how you should live your life and is there a God and if so which one.
I am Christian so will explain from my beliefs and views. I have come to the conclusion (my own view) even if you do not believe in God or a god, or even much of what is in the Bible; you can still be a Christian. I will try to explain. If you do believe in the Bible it is pretty clear Satan knows God is real, he was in heaven with him, so Satan must know for sure that God is for real. But it seems as though he will not be saved. So just believing that God is real is not going to save you. Basically the same logic goes for believing the bible, believing the stories will not save you either, if you believe the bible is true or not.
To be a Christian you need to have a set of values as defined by what Christ says in the New Testament; that is what a Christian is. Let me try to explain that. If you have a political view be it Socialism, Communism, or Capitalism. These are political views that can be defined, they are not really based on a person they are just a defined political view. If you say you are a Socialist, that is what you believe is the best government system and you would have faith in that system. I feel that to have faith or believe in Christs teaching makes you a Christian. I also feel you can be a Christian without knowing it. Just the same as you may have certain political views and not know what it is until someone gives you a definition. That is why I feel anyone can be a Christian and be saved as long as their beliefs are the same as Christs.
Now, for the doing good. I feel if you have the beliefs and values Christ teaches, you will follow this and do good and right. But making a mistake alone the way will not keep you out of heaven if good is in your heart. You mention Paul teaches faith and Christ teaches works. I feel they both teach the same, one may talk a bit more of one than the other. I think John 3:16 is all about faith. Also, David did some pretty bad things but God loved him, and the sinner on the cross was saved and never had a chance to do any good works. By saying this last part about the sinner on the cross, does not mean I feel just professing Jesus Christ will save you, I feel God can see into you heart and know what you believe for real.
That is a lot to try to get down in one post, so it may not be to well explained. I will try to do better if you have questions, or even if you feel I am a bit cracked in the head.