As for Amtrak you can take your bike from DC to Pittsburgh. It will have to be boxed. Very easy to do even at the station as boxes and tape are available. As long as a station has baggage service it can handle bikes.
The GAP has a few tunnels and one, Big Savage, is closed through the winter and is scheduled to open the first week of April. Check the Great Allegheny Passage FB page. Also get yourself a copy of the Trailbook. Well worth the cost.
I have ridden it a number of times and taken my bike from Cleveland to Pittsburgh and ridden to DC. This Spring my wife and I will be riding cross country starting in Delaware and going to DC toward Pittsburgh. We are planning to start about the first week in April so we hope the Big Savage tunnel is open.