Lots of options with folding bikes. The Capitol Ltd, as mentioned, makes stops along the route. See more info here. Stops along the C&O and GAP include Harper's Ferry, Cumberland, Md., Connellsville, Pa., and Pittsburgh.

I'm confused by the Syracuse leg. Is there some reason you are doing that? If not, then just hop on the Capitol Ltd. and head back to DC and then to FL.

Having ridden both, I think the GAP trail is more nicer and enjoyable to ride. If it were me, I'd ride the C&O from DC to Harper's Ferry (beware, though, the westbound train leaves around 11am), take the train to Cumberland, then ride to Connellsville. After Connellsville the GAP isn't quite as scenic as it is to the east.

You could also do a rental car as part of the leg. Hertz has a location in Cumberland not too far (1/4 mile) from the C&O/GAP trailheads.

If you are curious, I have my rides on the C&O and GAP written up on the Web at http://www.crazyguyonabike.com/briwasson