Can I do both Tiannemen Square and Badaling wall in ONE day? What would it cost?

Yes, of course. If you want to join a tour, I suggest you to contact TravelChinaGuide for help. They have similar route https://www.travelchinaguide.com/package/beijing-one-day-tours.htm and they can also help you to arrange your own trip. For the exact cost, you may need to ask them directly. I join their One day Beijing Tour including the Tiananmen, Great Wall and Ming Tombs, which me about $86. It is wonderful. They will provide you pick up and tour guide.
If you want travel on your own, you may cost about 300 to 350 RMB.