These are a lot of questions so I suggest that you ask them separately. I'll take your main one first. Hawthorne wants to emphasize the specialness almost supernatural quality about Pearl. What makes her so special? Well, she is born out of wedlock to a priest and a married woman. This exists so outside the draconian Puritan rules of "love and reproduction" that Pearl becomes special. Pearl is a child that represents freedom and unencumbered love. She is a product of a relationship that is human and real. Much like Hester, Pearl is a smart independent minded and gracious. Most women in this village are not. Check out this quote about Pearl,
"up and down, with the humorsome gesticulation of a little imp whose next freak might be to fly up the chimney...such was Pearl's wonderful intelligence that her mother half doubted whether she were not acquainted with the secret spell of her existence, and how she might reveal herself".
Pearl is like sunshine on a rainy day (yes I know bad cliché; it's all I can think of right now). She is larger than life, almost magical. She is a free spirit as opposed to the spirits that are shackled by constricting Puritan laws and beliefs of the village.