Tree frogs only take a few months to turn into adults so the kiddie pool shouldn't be too overtaken by algae and such in that time but things can be done to reduce the cesspool nature of the pond. You can add a very small pump to move the water a little bit. I have a tiny one in my 50 gallon pond that's less than 100 gph and runs on sunlight but I gave up on it because it only ran with direct sun and now doesn't run at all (clogged?). It's small enough to not bother the tadpoles in there.
To prevent mosquito larvae from multiplying, you can add products with Bt in them which is a bacteria that only kills certain insects. It won't harm the tadpoles at the proper dosage. Bt can be found in mosquito dunks, mosquito bits, and Microbe-Lift's liquid mosquito killer which I like because it doesn't leave brown bits of stuff (which tadpoles will chew on). See
http://www.fishpondinfo.com/mosquito.htm for details.
As for filters and pumps, you can try one of the little pumps which lack actual filters or perhaps use a sponge "filter" which is nothing but air bubbling through a sponge. Here's a sponge filter example:
http://www.drsfostersmith.com/product/prod...&pc=1&N=0&Nty=1Here's some small solar pumps. I bought mine elsewhere but I think it's the same as the smaller one.
http://www.drsfostersmith.com/Product/Prod...1&N=62728&Nty=1It's up to you if you want to add rocks to the bottom. It will make it more natural but you'll have to clean those out as well when you clean the pond.
I have a bunch of tub ponds, some with tadpoles. I only top them off as needed, add Bt, and totally clean them out once or twice a year. They do get kind of cesspooly but the animals don't mind. When fall arrives, I have to sort through all the water for the green frog tadpoles to move them to deeper ponds where they won't freeze solid.
You can do water changes on tub ponds or kiddie pools with tadpoles by using two buckets and a net. Get a bucket of water from the pond. Pour it through a net into another bucket. The net will catch the tadpoles who can be put back. Check the water that went through for any escapees and then dump it so new water can be added. If you have a lot of leaves and debris, you have to hand sift through it. I do that by dumping what's in the net on the ground and hand picking out tadpoles watching for their wiggling. It takes a lot of time, doesn't work for the tinier tadpoles, and a few will die from the rough handling which is why I only do that once a year.
You may want to consider replacing the kiddie pool with a more permanent pond for the frogs.