Okay. First of all, bettas need heaters. Even that much of a temperature swing will stress them. Plus, you need to heat your water before putting it in. And you didn't cycle your tank? That is a big mistake as well. Those large water changes, coupled with any ammonia/nitrites and cold water, a good equation for a stressed fish. And stress leads to sickness. Did you clean the gravel before putting it in? About the tank, despite what other members have said, 5 gallons is the recommended minimum. Anything below is just unhealthy. Try to get a bigger one when possible. You should also have more hiding places, such as plants. Mesh covers are inadequate because they allow water vapour escape. I also don't like the sound of chasing him around with a net, that will probably just frighten him. If you want to play with him, or give him something to do, put a sterilised ping-pong ball in for a while. Or put live foods in. Sorry I couldn't be more positive.
