Rivnuts from Homestead, PA on 7/29/2019 3:33:39 PM:
You've presented several questions in your post, some of which are intertwined. The question of how many miles per day you can ride is intertwined with your desire for "budget" lodging and, frankly, how much time you have to make the trip. With the training regimen you outlined between now and September, I would expect you could easily ride 30 to 45 miles per day in good weather. You may want to include in your itinerary a rest/layover day with no riding because of weather, trail conditions or need for additional rest.
The GAP rail trail and C&O Towpath consist of only modest and easily rideable grades along the route. Going from Pittsburgh to DC, you'll notice the modest uphill grade from Connellsville to Rockwood the most in my opinion but that section is still easily rideable albeit perhaps in one lower gear than you would use inroute to Connellsville. The "steepest" portion of the trip is on the GAP from the Eastern Continental Divide to Cumberland. However, going in the direction from Pittsburgh to DC, that section will be all downhill and a very easy ride.
I suggest you study the route map and its various guide books, including this bikecando.com website, and select/reserve lodging along the way that meets your definition of "budget". There are a couple of hostels along the way but your indoor lodging choices will primarily be motels, hotels and B&B's at varying costs and qualities.
While I can't do the entire trip in September, I'd be happy meet you in Pittsburgh and guide you out of town and ride down the trail on your first day ...assuming I am home that week. I live very near the trail just beyond downtown Pittsburgh and am very familiar with the city and trail.