After the conversion finishes, will PTS automatically do everything with the new PostgreSQL DB?
When it's finished converting open the File > Maintain Database Names... window and check the Dflt box next to the new PostgreSQL database. Then close and reopen the program.
Also, can I get rid of my 2nd (older) version of PostgreSQL?
Good question. I think you can but I'm not 100% sure. It would probably be safer to leave it.