Both me and my husband went to Penn, so I know the area pretty well. If you do live in the Old City, it will kinda suck to have to commute to Penn everyday. (Just because one of you has to, it does not mean you both do). Septa buses are not very good and the metro only takes you so far as Market and 30th: you would still have to walk 25 minutes to Wharton.
Check out Campus Apartments. They have plenty of buildings around campus, some not so great, some pretty nice. Ask me specifics when you start searching, I would love to help.
As for dog park, there is a park near Penn (38th Baltimore) which is not bad. That's what I used. It's got that West Philly feel where people stop and talk to you about your dog. Plus, most of the year there is a nice little Amish market there on Thursday afternoons and Saturdays.
Parking downtown is a nightmare, although the areas you mentioned are nice. Parking around Penn is actually not too bad, we did it all the time. Also, I think it will be much better for your husband to commute from Penn than downtown. He can talk the ramp to Jersey right from 30th Street (no traffic). (We are getting married in Princeton, so we did the Penn-Princeton route more than one time).
Seriously, consider living around Penn. It will be so much fun for you and equally easy for your husband. If I was not associated with Penn and did not need to commute, I would live in South Philly. Considering your school, the dog, the car: around Penn, I think, would be great. It's quite nice.