Hi!

Definitely get him to a vet. But in the meantime, here are a few things that might help. Get rid of the carpet. Use playsand, or newspaper, paper towels, corrugated packing paper. For some reason, I've never had good luck with the repti-carpet.

Even if he has a hot spot, the tank should be at least 85 degrees. My hot spots are at about 100 -115. Soak him in warm (not hot) water and not so deep he has so swim. Keep it warm, leave him in there about 20 mins. so he gets well hydrated. My beardies will not drink from a bowl, but always drink while in the bath, so that might help.

Baby beardies ususally do not eat veggies. They like things that move, so they can "hunt". You can try hand feeding: wax worms, phoenix worms, mini superworms, crickets, butter worms and dubia roaches all lightly dusted with calcium powder. Just stay away from mealworms, they really don't offer anything. More heat might stimulate his appetite, and certainly helps digestion.

I hope I have written something that you haven't already tried. Good Luck!