Manhattan is actually a wonderful place to run while seeing some cool sights.
1) Battery park is a great place to run, can get a bit crowded on weekends with tourists and street performers all over. Battery park is connected on both sides to jogging and cycling paths. The east side path goes through the South St Seaport and continues along the east river all the way to the North end of the island. Going along the West path offers very neat views and connects to a variety of other parks along the way. certainly recomended over the East side path.
2)Brooklyn Bridge run is fun and popular. Starting at the bridge its just over a mile to cross the bridge alone and another mile or so to prospect park. Probably about 5 miles total. Little bike lane to the park as well. Not a bad run, the bridge is a substantial hill.
3) No, south does no = downhill and north uphill. Youl encounter hills both ways and all races in CP do a loop. Going south on the east side has a large downhill at 70th st (cat hill) The west side is rolling hills the whole way. The North end of the park is a big vally, so either direction you go, you will encounter a large hill. If a CP race goes through here youl go down a big hill then back up another regardless of what way you go.
4) Yes, in central park. Many teams practice mid week there. Not sure what masters teams (student) but I see alot of em. Niketown also has group runs from the store at various times.
Have fun, running is the best way to really see Manhattan. Street running is not really as bad as people say