What emotions does Hawthorne himself seem to have about The Scarlett Letter?
These are all good questions; the last question is by far the easiest to answer.
The last paragraph in Chapter 19 creates a mood of melancholy reflection. It uses words to signal such a mood explicitly, like "fateful" (to realize something is fateful, you have to step back and think about it) and "melancholy." It also uses words that carry connotations of melancholy, like "whisper" or "dark."
As far as what Hawthorne felt about the story, that is a far more complex question. He worked on the novel for some time; one can assume he went through many emotions about it. We can't assume that these passages like the one in Chapter 19 are him thinking or feeling; those are his narrative voice (part of the fictional creation). To really understand what he felt, you'd have to step outside the novel and look into his journals, letters, and notes.
There is always the possibility of latent guilt regarding how people were treated during the Puritan times in America. Hawthorne's ancestors were from Salem and were very much involved in the witch burnings that took place there. In the novel, Hawthorne points out that people are not perfect...not even the most devout among the Puritans. He seems to be trying to warn us all about hasty judgments and shunning people for a single act or behavior.
Hawthorne shocked many with his choice of telling a tale of adulterous lovers. His emotion seems to be sympathetic toward the lovers. He portrays them with dignity and beauty, while the townspeople tend to be protrayed most unattractively. Hawthorne explores the nature of sin, and does not solely focus on the obvious sin of adultery.