It doesn't look like fin rot to me. If he's got any ripped fins it's probably because of any plastic plants in the tank, plastic plants are not something Betta fish should be with, their fins are very fragile.
Does the white thing look similar to these photos:
Sort of like a cauliflower growth on the tail? If so, it's very possible he has Lymphocystis, which is not curable but can be contagious to other fish if they happen to have open wounds or a deficient immune system.
If you could be so kind as to copy and paste this to your reply and fill out as many questions as possible, it would better help us help you out.
What size is your tank?
What temperature is your tank?
Does your tank have a filter?
Does your tank have an air stone or other type of aeration?
Is your tank heated?
What tank mates does your betta fish live with?
What type of food do you feed your betta fish?
How often do you feed your betta fish?
How often do you perform a water change?
What percentage of the water do you change when you perform a water change?
What type of additives do you add to the water when you perform a water change?
Have you tested your water? If so, what are the following parameters?
Symptoms and Treatment
How has your betta fish's appearance changed?
How has your betta fish's behavior changed?
When did you start noticing the symptoms?
Have you started treating your fish? If so, how?
Does your fish have any history of being ill?
How old is your fish (approximately)?
Do any of these other photos look similar to what your fish is going through?
Hopefully these photos show up for you, if not I can direct to elsewhere to see these images.