I'm a bit north of Philly (closer to Trenton, NJ) but here's some answers that might be of interest:
1. You really start to feel fall weather toward the end of September/beginning of October. You start feeling spring weather in April. During the winter, the weather also fluxuates - some days can be really cold and others be spring like.
2. Most of the time, you get snow in January and February with chances of it starting in December and going on through March. This year, we had snow in my area at the end of October, a few days before Halloween. (Trust me, I was gripping about it! Snow shouldn't come that early!) We've also had a freak snow shower in April several years back. (I remember that because it was the weekend a friend of a co-worker got married.)
3. I think Philly itself would be an 'attraction', given its history. But here are some things I know are "in the area". Most of this depends on whether or not she has a car/can drive or can get someone to drive her.
- Pennsbury Manor - in Bristol, PA. It's a historical site, William Penn's 'summer' home. (I believe it has a website.)
- New Hope, PA/Lambertville, NJ - these are 'tourist' traps/artist colonies. Essentially cute little towns with a lot of antique/trendy shops catering to tourists. If she's there enough, she might get to know some of the locals/shop owners.
- Princeton - it's got a really good regional theatre (McCarter's) as well as being a University town which might be a big draw for a teenager. It also has several Rev War sites if she's interested in historical stuff but I would think hanging out with the college students would be a bigger draw. And it has some cute little shops too. Suggestion: Small World Cafe is a popular coffee shop in town.
4. Don't really have any answers for this one
5. Depending on what you want done with the school aspect of things but since there's a lot of Revolutionary War sites (Washington Crossing/Philly) and other historical sites (Pennsbury Manor [colonial period] and Howell Living History Farm [1890-1910 farm that's an actual working farm using historical techniques]), your MC might go field trips to these sites. They do offer tours for classes. They also have websites you might want to look into.
The stuff I mentioned are all located up the Delaware River. Another option is going into South Jersey (i.e. the Pine Barrens) or into Camden (not recommended unless one of the other character's from there) or, most likely, out to Atlantic City.
Another thing - she might hear something particularly Jersey. People in NJ say "down the shore" when going to the Jersey shore. It doesn't matter which direction you head, north or south. It's still "down the shore". Another thing you might want to look into is a magazine called "Weird NJ". That offers "weird" locations as well as interesting personalities.