Who is the father of Jen's baby?
The show never explicitly reveals who the father is. It could be Billy.
What happened to Bodie (Bessie's boyfriend)?
He turns up in a few of the seasons, you definitely see him in season 3 and 4. He eventually was said to have taken a job as a chef on a cruise ship to support Bessie & their son, which explained his lengthy absences from Capeside.
This was filmed in Wilmington, North Carolina at Screen Gems Studio.