Don't worry, you did a good thing by saving him.
Here are some tips:
- a 5 gallon should be changed 50% every week, using a gravel vacuum
, suck up the water at the bottom where the poop/gross decaying stuff is.
- bettas excrete ammonia (its like pee) which builds up in a tank, if it becomes testable (greater than 0) than it starts to burn/hurt the fish so if you have a ammonia test kit, test every week before a water change :)
- a filter creates place for good bacteria to live, this is called the nitrogen cycle, it can be confusing, you can read about it here
but I wouldn't worry about it until you have caring for bettas down! so just ignore it for now. ^^
- since your tank is not cycled, if you feel like the filter is strong, turn it off for now.
- its okay if you added the bettafix, try doing a 50% water change.
Tail biting is really common, sometimes they do for it now reason so don't sweat it. as long as the water is ammonia/nirite free, it won't get infected. here is what I think happened, you haven't changed the water for 3 weeks so there was ammonia/nitrite built up which stressed your fish that may have triggered tail biting. as long as you change the water on time and keep the tank water clean, he might stop biting.
do you have a heater? warm water can really help boost their immune system. :)