Agree, definitely a victim of poor water quality and cold temperatures. I know that in pet stores they tell you that bettas can live happily in small tanks with no heater, no filter, ect. That is completely false. Bettas are happiest in tanks 2.5gallons+ with heaters and slow moving filters.
A 1 gallon tank is hard to heat, the temperature needs to be monitored closely. Your betta at this point needs regular water changes, every day to every other day, the water temp needs to be between 76-82 degrees F. And aquarium salt is a good idea!
On a side note, your betta is definitely a male, not a female!
Good luck! If you have any other questions please dont hesitate to ask!!