The new terminal at NKG is certainly a step up from the old one.
Just off the top of my head here are some answers to your questions.
Yes, you would clear immigration in PVG. Sometimes, depending where you are coming from, your bags will go to the destination. Make sure you ask that question upon check-in and again when you get to PVG. I have had that happen from Taiwan. It was still a rather confusing walk to get from the international arrival to the domestic departure. It is no where near as clear as it would be in a standard U.S. international airport. If in doubt, ask.
I'm Delta Platinum Medallion. China Eastern and others in China affiliated with SkyTeam are pretty good about recognizing your status. The lounges are nothing to write home about though.
Getting from the airport to the city is a long drive, about 45 minutes to an hour. Taxis are pretty good, but it would be good to have your destination written in Chinese to show to the driver (print out hotel address or something before leaving).
There is a lot to see and do in Nanjing. In no particular order: the Nanjing Massacre Museum is well worth a visit if you know about that part of WWII history; the old city walls and gates are pretty cool. There is a good reconstructed section open to tourists; Dr. Sun Yat Sen's Mausoleum and the Ming tombs are on a mountain outside of the city but within easy cab ride. It's a nice scenic area to visit, but there are a lot of stairs; the Xuanwu Lake park right outside the old city wall is nice place to go for a stroll.
If you have time and would like to see one of the best places in China go to Yangzhou to see Slender West Lake. Also outside the north gate is Da Ming Temple and a short walk to the east is Songjia Town relic park, which is an old Song Dynasty fort and grounds.