Hawthorne saw secret sin as something that separates people--from God and from each other. By refusing to adknowledge their sins, Hawthorne's characters doom themselves to lives of isolation.
Arthur's efforts to "come clean" are not honest; he still can't bring himself to simply say what he had done. Consequently, when the congregation responds by deeming him to be even holier, Arthur is even more isolated from the community.
In Chapter 11 Arthur is moving closer to real confession and redemption, but he isn't there yet. When he finally does rid himself of secret sin by standing on the scaffold with Hester and Pearl in the light of day (instead of the middle of the night), he is no longer estranged from God and the human community.
In response to #2 - No one burned in the Salem Witch Trials. 14 women and 5 men were hanged, 1 man was pressed to death. The Witch Hysteria in the 1690's had John Hathorne as one of the hanging judges, and the only one who did not eventually recant his actions; Nathaniel Hawthorne, living in the 1800's, changed his surname to distance himnself from his unrepentant ancestor. See http://www.iath.virginia.edu/salem/people/j_hathorne.html
Many a Puritan endured secret sin because the consequences of confessing sin were prison, ostracism, and degradation and even death. The sad fact is that the Puritans, in their zeal to "purify" went overboard. For, these zealots denied people their humanity. Afterall, it is human for people to sin.
And so, in Chapter 11 Dimmesdale with his guilt and secret sin becomes much more human and less "ethereal" as the townspeople perceived him in Chapter 4 as his own suffering enables him to sympathize with the sin and suffering of others. The paradox here is that the more that Dimmesdale asserts his own sinfulness, the more the townspeople elevate him in holiness. Realizing this, Dimmesdale feels even more the hypocrite. His sin is compounded by his subtle attempts to confess; his terrible guilt leads him to scourge himself and keep all-night vigils, denying himself rest.
Chillingworth who, as Hester later declares, has "violated the sanctity of the human heart" appears blacker than ever; he is "more wretched than his victim" into whose heart he delves with a vengeance.
Through the characters of Dimmesdale and Chillingworth, Hawthorne exposes secret sin as more destructive and evil than the overt, confessed sin of Hester. While Hester has paid the price of sin with ostracism and loss of her beauty, she has been forgiven by some in the community and is seen as a charitable figure. But, Chillingworth, the worst sinner of all, grows blacker, more deformed, and fiendish. Chapter 11 furthers the development of Hawthorne's theme of not hiding sin stated in the conclusion as the moral lesson to be learned from the "minister's miserable experience": "Be true, Be true! Be true! Show freely to the world, if not your worst, yet some trait by which the worst may be inferred!"
Throughout the book and in this chapter, Hawthorne is suggesting that the more you hide a sin, the more it haunts and affects you. The only way to truly get the forgiveness we hope for and deserve, we must face it, admit it, work through it toward redemption as Hester does. He was probably suffering from this a little himself since his ancestors were involved in the orginal Salem witch trials and responsible for the hanging/burning of those innocent people.