If you have a cat go with a large rabbit cage, that way he has a secure cover and they are not to expensive. For substrate try aspen or bead a beast/sand mix. I personally use aspen for my less humid torts/turts, then I place a humid hide in there and a small section of bed a beast for a little humidity. I agree with Robyn either get a really small water dish (I use small plant saucers for the little ones) or remove it when you are not home.

I think, I would probably go with a rabbit cage. ( I just got one after having my adventure outside with Comet. this way he will be safe inside or out). The ones with the deep plastic bottoms would work nice I just picked a new one up at Petsmart for about $25.00 but you can often find them cheaper on Craigs list or in thrift stores. Or if you are handy with hammer, saw, and nails you could use a plastic tote and make cover over it. But remember UVB does not penetrate through screen well. They do have the 1/2 inch mesh screen (uvb should penetrate ok) at home depot that you could make a cover out of it.

                Here is a picture of where we have him. He is very active when he is awake, walking and climbing. He has this little frog like action going on under his chin sometimes when he is basking in the sun, is this normal?

A rabbit cage might be a good\safe solution.

Most of mine make that movement too. Nina, you have made a very pleasant looking habitat for your Leo but it has a few drawbacks. UVB light needs to be without a barrier, even a screen can cause some of the uvb light to not reach its intended destination. What kind and size uvb are you using? See site below.
http://www.uvguide.co.uk/fluorescenttubemeshtests.htm
There is a thread that deals with bad lighting. It is an interesting read.
I would change the substrate to aspen or play sand / bed a beast (coco fiber) and/or dirt mixture.
Take out the large things that cause him to flip for now. I would place him in a rabbit cage if you want use your backdrop to line around the cage to help hold some heat and moisture in. Keep a check on these if they get too high take it away. The water bowl you did great when you placed rock inside it. That way he can get a drink but not flip. Just make sure if he can get over the side it does not cause him to flip. Or use a smaller bowl or remove it when you are not home. It takes us all time to tweak the environments of our shelled friends. I do not have Leo’s so I am sure someone who owns will chime in and add comments too. Keep up the good work your on the right track.