I own a 5.5 gallon tank. This is probably the smallest size I would recommend for a single male betta. His tank mate is a single ivory mystery snail, and I have tried ghost shrimp. Deathwing ate the ghost shrimp.
I plan on upgrading to a 10 gallon tank within the next few months. In my short and limited experience, buying small always ends up with you spending more money. I would suggest going no smaller than a 10 gallon. Currently, I have an AquaClear/Fluval 30 hang on back (HOB) filter. Works great. The intake pipe is such that you don't have to worry about your fish getting sucked up. If you are still worried about it, they make "baffles" that fit around the intake tube usually used in tanks with small fry/shrimp. I would not purchase any filter that doesn't have adjustable flow. I don't remember why, but through my betta research, I recall more experienced fish keepers' advice to stay away from under gravel filters. Bettas don't care much for strong currents, so I have waterfall portion of my filter baffled with an extra sponge. Keeps the current down since I have such a powerful filter on a small tank. You should probably aim for a 75w heater. Bettas like warm water. I'd also avoid the pre-packaged starter tanks. They seem like a savings in the beginning, but as you start learning more about tanks and fish, you find out that the filters/heaters/etc. they package are not sufficient enough for a healthy tank. I have replaced everything that came with my 5.5g starter kit. Had I just bought the items I have now with a glass 10g, I would have saved close to $100.00.
I have also heard/read of people keeping African dwarf frogs with their bettas successfully.