Well the other container won't be cycled either so you are better off just leaving the fish in a doing a 25% water change every week (using the fish to start the cycle). You should really have done a fishless cycle before buying the fish.
1. 20 gallons for a single fancy minimum or 50 gallons for a single comet/common minimum.
2. NO
3.bettas need a filter heater and 5+ gallon aquarium but plants and gravel are optional (but would be prefered.)
Why was your daughter trying to catch the fish? You dont need to take the fish out during weekly 25% water changes. As long as he is in a proper humane environment (5+ gallon heated and filtered aquarium) his fin should grow back.