I would not use melafix since I have known it to have bad side effects for bettas. The main ingredient melaluca can actually effect their labrynth organ...I remember there was a link floating around somewhere a while ago to a good article on that but I don't remember where I got it.
For a 2.3 gallon tank I'm thinking you'll have to do 100% water changes at least twice a week. At first you'll have to test a lot to get it right. The 100% water changes aren't stressful if done right (water is from the same source and the same temperature) and are necessary for an uncycled tank.
As for the rip in the fins, I would use a dose of aquarium salt (probably 1 tsp for a tank that size) and see if it will heal on it's own. If it should get worse then I recommend Jungle Fungus Clear. Just keep in mind that bad water quality is the worst thing that can happen, so you may have to modify the med schedule if it calls for less frequent water changes than your tank needs.
As for the filter, it won't help unless the tank is cycled. I would hold off using it until you get a bigger tank and if it is too strong you can baffle it. Murky/cloudy water means you need a water change, but the water should really be changed before it gets cloudy.
I also agree that you should raise the temp a few degrees to at least 78. I keep mine all at 80 and they love it.
I hope this helps ;)