Do not use melafix. It coats their labrynth organ and cause them to suffocate. You need to do a 100% change to remove it.
Fin rot is caused by water quality issues, just like the slime coat shed can be. I would add an extra 50% change mid week from now on. Also while he is healing from fin rot I would do 50% changes every day or at least every other day along with the weekly 100% change. Otherwise I would do aquarium salt predissolved at 1 tsp per gallon for the rot. You only add more salt with a water change and then only add as much salt as water you change. You can do this for up to 10 days. If that doesn't work and it seems to keep progressing despite the salt you can try an antibiotic like Kanplex, Triple Sulfa or Furan 2.
He also needs a heater and an intank thermometer to monitor temp. you can get a 25w adjustable heater like Marineland Visitherm or Jager, which would be ideal. You also have to use an in tank thermometer with the heater to make sure it is working and set it properly (don't rely on the heater). I also highly suggest testing your heater for 24 hours with thermometer in similar size container to make sure it will hold an appropriate constant temperature. Once you are sure it is working your betta must be acclimated to warmer water slowly at no more than 1 degree per hour and 5 degrees per day, or floating in a plastic cup (solo type) in the bowl for an hour, after the water has already been fully warmed.