For the driver, 1000 isn't bad but its not the cheapest. I think I paid 700-800 for a full day (8am-9pm).
I would do the wall in the morning as it will be less crowded. As mentioned Badaling is the most visited so you may want to avoid it.
Mutianyu is much less crowded and a nice stretch. (Wear comfy shoes, its quite the hike)
Tiananmen Square is just a massive square, nothing much to do except walk around.
Forbidden City is just across the road from the square.
I found it very interesting but some do find it just more of the same thing (it kind of is. Amazing how many rooms can be used
for one persons needs)
Summer Palace was also nice. Its a bit of a drive IIRC but you can probably do it in the early afternoon.
Olympic Park is cool, even just for a pic with the Birds Nest Stadium and the Cube swimming center if that interests you. But probably not something id do if pressed for time.
There are tons of markets. Don't think you'll get to the night one. Silk Market is great for real fakes. There's also the Pearl market and the antique market amongst others.
Kung Fu show was incredible but with leaving at 8, don't think you'll make it.
Dini's is good food and the Rabbi and Rebbetzin are really nice!
If you're so inclined, you can go for shachris and have breakfast there afterwards.
Just leave time for things as in the city there can be traffic. I'm sure the driver can help you with that.
I really shouldve written a TR...
If you have any questions, please ask and I will answer to the best of my abilities.
And pray for a clear day...
Photo from Seth MillerAnd what it really looks like,