Water changes must be done MUCH more frequently. For a tank of that size, perhaps two 25% or one 50% water change per week. I seriously recommend you doinga 100% water change immediately, clean any ornaments in his tank, and rinse any gravel with water of the same temperature as that kept in the tank to not destroy any beneficial bacteria (basically just to clean the gravel of any poop, leftover food, etc.)
I HIGHLY recommend you get a heater, especially with winter coming up. He should be kept at at least 80 degrees, 70 is way too cold.
Also, do you have plastic plants in his tank? They could be partly to blame for the torn fins. Anything with pointed leaves or bits of excess plastic sticking out can tear betta fish's fins. I suggest taking them out if you have them.
If the case of fin rot, if salt baths don't help, I'd recommend treating it with Maracyn + Maracyn-two. Do a full week of treatment, preferably in a hospital tank.
What I did with my 3 gallon was fill it with 1 gallon of water without any gravel or plants and only left a little cave in to give my betta some shelter (helps relieve stress). Dissolved 1 teaspoon of aquarium salt in the 1 gallon (before I put the fish in), and I put like a pinch of Maracyn and Maracyn-two every day for 5 days. I also added 1/2 a teaspoon of bettafix every day to helped promote fin regrowth, but I don't know if it really helped or not. After the 5th day, do a 100% water change, and wait two days without adding any medicine and if the fin rot acts up again, do another week of treatment.
Also, it sounds like he might possibly have air bubble disease.
[edit:] OH MAN, how could you let it get that bad?! No wonder he has a hard time swimming! If you let his tail completely rot away, he's going to drown (YES he can drown! Betta fish have a special organ that lets them breath oxygen, and they NEED to breath air)! Lower your water levels ASAP to help him be able to reach the water surface easier. Forget salt baths, go to medicine NOW. ): It looks like something my friend's fish had, if he still has any salvageable finage he might be able to make it, otherwise, if his fin looks like a swollen stump, you'll have no luck growing it back ):