I think you could get an african dwarf frog in a 3 gallon but I am not 100% sure as I have never kept them before. I heard they can escape through the tiniest holes so make sure you cover everything in your tank that they can possibly escape through, even if it is way smaller than them.
I am not sure of what african dwarf frogs eat either.
I'm sure they would probably eat sinking pellets but I am not 100% sure of that.
I would not get livebearers, even 2-3 males will nip at each other and harass each other in a 3 gallon. I had 2 male guppies in a 10 gallon and the larger one killed the smaller one and ripped it's fins to shreds.
I think the plant idea would be very cool! It would look very nice with a plant on each side. Some good hardy plants are java fern, java moss, and moneywort. I keep moneywort in my snail's tank that doesn't have a light on it. It just gets light from the light in my room and the light next to it on the table and it grows fine.
I think you're limited to only a male betta, a female betta, or an african dwarf frog. There are dwarf species out there, like dwarf cories, but they like to be in groups so I would go for the male betta, the female betta, or the african dwarf frog.
What about some shrimp?
About the heater: You should invest in a heater especially if it gets cold where you live in the winter and hotter in the summer.
You'd need the heater to keep the temperature constant so it isn't always changing. You should be able to pick up a small heater for like $5.00. I bought a small heater for my 10 gallon and it was only $8.00. Make sure you buy a thermometer to put in the tank as well so you can always see what the temperature is at.