I've ridden it in September, and it was quite cool at the top (Big Meadow). Big Meadow is 50 miles from Front Royal, so it looks like you won't be going that far (I rode up there, spent the night in the lodge, and rode back down. "Down" is relative, since there's nearly as much climbing going back to Front Royal as there is going up.)

The lower elevations are nicely shaded. There are sunny stretches as you get higher. But in general, it'll be cooler the higher you go. There are several places to pull off and sit in the shade at a picnic table, and at least one store.

I don't recall my speed, but I was climbing in the 5-6mph range. Dark chocolate M&Ms helped alot. The first 5-6 miles is a pretty steady climb and then it goes up and down. There is one tunnel to be aware of--it's a good idea to have a blinkie and a headlight if you are going to go through the tunnel (uphill on the way up, downhill on the way back down). I just had to surrender to The Force going through that tunnel since I had neither. It was rather nerve-wracking.

I would suggest not fretting about your speed, and instead just enjoy the ride. I've taken this advice to heart and removed my computer on some recent rides and it's a really positive experience.

Have a good time. There's a 7-11 at the bottom near the entrance if you need any last-minute treats.