I would hold out and wait to see if the natural bateria colony forms to clear the water before trying to put in chemicals...many of the chemicals just don't work that great anyway, especially anything that promises to break down and reduce organic waste.
Many people feed hatchlings daily, and then once they hit juv. size start transitioning to every other day. Once they are an adult, definitely every other day. Many that feed adults on that schedule do chose to offer some veggies on a daily basis, and restrict the Reptomin (or similar protein sources) to just every other day, or a touch less.
You might want to break out that Reptilite bulb...to the best of my understanding, UVB light doesn't pass through glass and acrylic surfaces like windows and tank walls. Meaning that you either have to use a bulb that specifically produces UVB (not just UV) rays or provide some outdoor time in the real sunshine. (A nice Rubbermaid with a shaded corner and a basking spot should do it, and keep it cheap.) 15-30 mins. 1-2 times a week should do it. I personally sprung for the bulb (one reptile heat bulb for his basking spot, and the uvb bulb which is a tube-style.) I live in an apartment so getting the bulb was easier that carting him down the stairs to the yard.
And yes, your tank size is ok for now, but you'll want to start saving up for the great big "rest of his lifetime" tank. The thing about having a foot still touch the bottom with the head out of water is completely false. RES are strong swimmers, and I can't imagine a wild one would limit what ponds and streams he lived in because he didn't have a foot touching when he stuck his head out to breath. When you think of it in the terms of what a wild RES would do, such a guideline sounds very silly.
You've got a cute little squirt...best of luck!