Yes. The train allows bikes with a reservation. IIRC, only 5/train. There are two hotels within a stone’s throw of the PGH station.

IME, Avis has a lot of one-way rental options. Don’t assume that you have to drop off at the airport. The first time I did the GAP one way I rented a car in Philadelphia, drove to PGH, stayed in a hotel and dropped off the car near Pitt. From there, I rode to the start of the GAP.