Well, me myself wouldnt put the betta with the frogs. im glad you would put the frogs in a bigger tank but they are air breathers and they will come up for air, I have heard several sources say they need a low tank, as in the 10 gallon, so they dont struggle to swim up and up to get a gulp of air. You could get the 20 gallon, and if you really want to get something else besides your little girl, you could get guppies, they are fancy, colourful and crazy. If you want you can get both genders but they are called million fish for a reason, so you better get 2-3 females per male or get only males. I would recommend you getting the guppies first and putting them in first as well and leaving them in the tank for a while and then put in the betta so she doesnt think later on shes the Queen of the whole tank and chase them around.
As for cycling, Im not an expert in explaining it but there are bacteria called Nitrosomas and these convert ammonia (produced by fishes respiration and waste as well as rotting food etc) into nitrite, both NH3 and NO2 are toxic, which then nitrite (NO2) is being converted, by a different kind of bacteria, into nitrate (NO3). Being it, all three are toxic and/or deadly to fish where nitrate being toxic only above 20ppm and IF fish are exposed to it long term, it can be deadly or will have an impact on their reproductive organs. Also, pH of the water the fish are in plays a huge part on how toxic the ammonia is the higher the pH is the more tocic ammonia is. Ammonia can cause poisoning in fish as much as nitrite can, these usually are deadly to fish, and NH3 will cause a decrease in O2 in H2 and will cause the fish to gasp on water surface or cause them to have pink/red ish gills.
There are two types of cycling your tank, fish in and fishless. What to expect, fish less cycle is without fish in the tank, you either add pure ammonia (like Dr Tims one) or fish flakes in and leave them to rot there and release ammonia for bacteria to feast on and you check water for ammonia and nitrite only, daily, until you see both on 0ppm daily without crashing with a sudden rise in nitrates and then after fish less cycle you put the fish in so the cycle doesnt stall or crash. Fish in cycle is when you cycle your tank with fish in the tank, you do water changes of 40-50% daily or every two days with testing for ammonia and nitrite daily, until you see them both on 0ppm daily without crashing with a sudden rise in nitrates.
When there is a lack of oxygen or ammonia source in water, bacteria go dormant which doesnt mean you still have them, individuals will die off.