So, distilled water is the best to use for my red eye tree frog's misting or fogger, right?

Yes, only because it doesn't leave much residue from dissolved solids on the front of the enclosure and it won't clog the misting nozzles as quickly.

And it's best to buy distilled water from a store because I don't have distillation set up at my house, right?

Yes, it is very cheap and easy to keep 4 or 5 gallons on hand.
And dechlorinated water without all the minerals removed is best for the water dish, right?

Yes, you don't want aggressive water for them to soak in.

So, would tap water put through a carbon filter suffice; or, is municipal tap water treated with a de-chlorinating agent be better?

Personally I would treat the tap water. If you use a carbon filter you will not know when breakthrough occurs and I wouldn't use anything that I can't be sure of, and is testable.

Thanks in advance for the clarification [/QUOTE]