Agree on temp, bettas are tropical fish and require warmth to stay healthy.
10 gallons, unfiltered you would need to be doing 1 50% water change per week and 1 100% water change with gravel cleaning per month. A filter will eliminate the need for the 100%, but you would continue to do 50% per week with regular vacuuming of the substrate.
If the bettas are the only fish in the tank then an air stone is not needed as they don't normally breathe through their gills- they get their oxygen through the air at the surface using a labyrinth organ. If the bubble are too strong it could disrupt the water's surface and make it harder for them to get the air properly.
If you have other types of fish in the tank, highly suggest using a filter since a cycled tank is needed with most other fish and not dividing by half as most other fish need the whole 10 gallons and then some- why bettas are popular for small tanks, since they are better by themselves, smaller in size, and don't require a cycled tank.
As for the fin- most likely it was damaged during shipping, a net was used when collecting him, or a birth defect.
If you see no signs of trauma on his fin, or immediately around it, then I would wait and see how he does in time.. as there really is nothing you can do for it to fix it at this point.