From what I've learned with my betta and from all these experiences on the forum, bettas can NOT do well in small unheated, unfiltered tanks. The low temp (they need temps in the high 70s all the time) makes them vulnerable to illness and without a filter dangerous ammonia levels can occur.
I would strongly recommend upgrading to at least a 3 gallon tank with a heater and filter and treating him for his illness. Also when you do water changes make sure the temperature of the new water is as close as possible to the water in the tank or your fish can go into shock.
Also you said your friend told you not to use chemicals - he's wrong, if you don't leave your water out for at least 24 hours before putting it in the tank you must use water conditioner to get rid of the chlorine which can poison your fish.
I hope your fish gets better!