Help with Budgett frog. Brand new

Hello all, glad I found this forum. I've been looking for a while now for people who have first hand experience with these guys. I've been wanting a budgett frog for months now and I just got one yesterday and I'm super excited. It's an adult and I have him in a ten gal tank with about 1/3 of the tank filled with water(is that too low?). I have a filter in it and some flat stones for him to get on to get out of the water. I have a cheapo heater that does a poor job and I live in Washington where it can get pretty cold at night and we don't heat the house, only the room we sleep in. So the guy told me to always have a redlight on, on top of the frog's tank to help with heat. I was thinking about just getting a better heater to keep the heat up but he said just do the light. But it is on all the time, even at night, and I think it might be stressing the frog out. Should I just get a better aquarium heater and ditch the whole redight thing? Should I worry about the frog burning himself on the heater? And I'm guessing the water temp should be at like 78 F degrees? Then, I know I just got him yesterday, but how long should he go without eating? I tried feeding him an earthworm yesterday and today and he is not interested. I've only had him for 2 days, but I always worry when my pets don't eat. Any and all help would be greatly appreciated! Super excited about this guy but also a bit worried. I read somewhere(typical internet research results) that these guys can die easily, so I'm a bit spooked and nervous. Again, thanks for the help and I would appreciate any help as soon as possible. Today I have a free day to go out and get what is needed to ensure I have a great frog long term. Thanks!!!