Sorry this is so long!!
Thanks for the pic, Thunder 22. It looks to me like he is biting his tail (my guy did that for days when he first came home and it looks similar, I think. Pic below. He didn't bite off as much as your guy has, but I think it's what's happening.
Search the forum for some threads on tail biting. They do it for all kinds of reasons. Sometimes it's from stress, sometimes their tails are too heavy (especially if they're tiny) and sometimes it's just a habit without any apparent cause. Since you have had your fish for a while and he just started, I think it's probably not habit and that there is something stressing him out about his environment.
Which brings us back to the water temperature and conditioning. I think changing those things will solve both your problems - it will perk him up and he should stop biting his tail. If you do these things and his tail still has problems, we'll deal with it then, but these are the first steps and are likely to fix everything.
Don't use distilled water, as turtle10 says. Unless there is something wrong with your tap water, you should use it because it has beneficial minerals for your fish. It just also has harmful chemicals, and using a water conditioner will remove the harmful chemicals without removing the good stuff. There are a lot of water conditioners around that are perfectly fine, though a lot of us recommend Seachem PRIME. I think it is better at combating ammonia, and it is SUPER concentrated so you only use two drops per gallon and it lasts you forever. It also doesn't cloud the water like some conditioners do. Here's my recommendation:
Small heater suitable for 1 gal bowl
Seachem PRIME conditioner
Thermometer (if you don't have one already)
API Liquid Test Kit
Prepare fishy's water the night before in a 1 gal container. Fill it with tap water, add 2 drops of PRIME (from a small dropper) and 1tsp Aquarium Salt. I have a second heater that I use to warm the water to exactly the same temperature as in his tank, but you have to make sure that your container is heat safe, and not everyone can get a second heater. If you can, it works very well for me. If not, just let the container with PRIME and aq. salt stand overnight. The next night, test the water parameters with your liquid test kit and make sure there is no evidence of ammonia, nitrites or nitrates, that the water is not too hard or too soft, and that your pH is between 6.8 and 7.2. Then remove fishy and perform 100% water change. After adding the new water, add your heater and thermometer. When everything is arranged, add him back in the tank.
You want him to go back into the tank before the water is heated so he doesn't get heat shock. The water will warm slowly with him in it. Monitor his temperature every day. Even a small heater will sometimes warm the water too much in a 1 gal tank, and you may want to turn it off for a while.
Repeat your water changes every night, and include the 1tsp Aquarium salt for 10 days for his fins. After 10 days stop the salt but continue the PRIME always. If he continues biting, you see black edges, or his fins don't start to grow back within a couple weeks, come post in the Betta Diseases and Emergencies section.
Let us know how it goes!