Can you catch a taxi from Beijing airport to Great Wall (Badaling) and catch one back to airport?

Yes, you can take a taxi for a round trip. But you need to inform the driver to wait you back, since there are few taxies in the Great Wall. It costs CNY250 or around for a single trip.

By the way, I was told TravelChinaGuide provides the service. They will arrange a good car and English speaking tour guide for you. You may contact them through booking travelchinaguide.com