A 1/2G bowl is way too small for a betta. They need at least 2Gs or more. In a bowl that small, his fins won't be able to grow that long and he's probably shredding his tail because he's agitated. Also, you can't safely heat a 1/2G. They're warm water fish and need to be in 76-80 degrees F. They also love hiding in plants and hiding areas (you can make them or buy them). I haven't dealt with ick so I can't help you there.
I hope you don't mind, but I'm going to show you a picture of one of my VT's Valentino. He was in a VERY small bowl when I bought him and I've had him in a 2G tank for a couple months now and his tail just keeps growing. This just shows that more space makes them happier and makes their tails grow more.
Some people think that smaller tanks are cheaper than bigger ones and that's not always the case. I bought this 2G no-name at Petco for $20 and I love it!