That would be towards the end of hurricane season, although this year seems pretty active so you could have copious rain (and mud
) Could be cold, could be hot.
Camping along the GAP is pretty much limited to established campgrounds and state parks, with 10 - 15 miles between them. Most have full services available. The C&O offers free hiker/biker sites every 5-8 miles with water and port-a-potties, in addition to commercial camps near towns. So a little more planning for the GAP, but I shouldn't think that you'll need reservations.
I rode all of the C&O and about half of the GAP and was able to buy food every day. Water at the hiker bikers below mile 30 on the C&O was an issue. If a pump fails the weekly water quality test the NPS removes the pump handle. All of the hiker bikers between Georgetown and mile 30 were without handles. I also flew by the seat of my pants as far as camping and food stops.
The surface of the GAP is general smoother than the C&O, in my opinion the worst of the GAP is about the same as the average for the C&O. Both have rough sections.
I rode on 700cx35 (37) cyclocross tires and they were fine - those would equate to just over 1.25 in width. A little tread helps keep the tire from sliding sideways along the edge of puddles.
You should see less traffic than I did, by then you may have a lot of alone time during the week.
Good Luck and Good Journey
Doug