Yes and yes. Faith is partly up to us, and we have the power to deny it. Sin -- the reality of which is amply demonstrated by Scripture -- is us turning away from God. If we have the power to turn away, we obviously have the power to turn to Him, as well.
I suppose one could argue that sin is God's will, but that seems a bit strange.
If we could actually deny God, then God would no longer be sovereign. I find Protestants to be confused by two events. One is their salvation. Scripture tells us that the Son must set us free and, if we are freed then we will be free indeed. Now one has to ask what precisely are we being set free from-especially if we have it within our power to free ourselves? What does the Holy Spirit do?
The second event is after we become Christians. While we certainly now are free to choose to follow God or follow our own ways, what is the purpose of God "chastening" us. Why wasn't Jonah free to go to Tarshish or Paul to Damascus?
Some Christians seem to think in terms of either-or. "If God allows sin then He must be the author of it." Rather think of it that God does all good things and uses evil to effect His perfect plan. Everything evil could be gone in the blink of an eye if God so ordained it. However, if one views God as using the evil in this universe to carry out His perfect and righteous plan, then we know that all things work together for our good. Every thing that is evil that comes into our lives is allowed by God simply because it benefits and blesses us.