1.) I use a reptitemp 500r atm also, and that happens also. It is what happens with it. ITs basically a glorified dimmer switch..does not stay at an exact temp if the room temperature changes a bit. 90-93 is a good temp so I honestly wouldnt worry much. Now if it was dropping below 90 , I would worry. Mine stays between 90-94.

2.) You should have atleast one water bowl in there for them to drink from, even if it is small. Does your tank have a screen lid? Or are you using a tub? What substrate are you using?