I use NLS baby fish formula. Mine seem to eat it without a problem!! Otherwise I crush adult food.
If you go with a filter, I recommend finding a sponge filter. That will be gentler for the fry... Plus even with adult bettas it works nicely
(sponge filter, pump, air hose, and adjustable valve).
What is most important though, is a heater. The water should be at 78-80 degrees, especially for a poor little betta who is probably stunted, burned by ammonia, and cold (not because of you, but because of being sold young).
If she is new, she may not eat yet. Try again tomorrow. If you can, find live foods. Most cannot resist live foods.
2-5 gallons is up to you... Whatever works for you, and as long as you can find an adjustable heater to fit in it, you're good.