Do you know how much water the tank holds? A single betta needs at least 2.5 gallons or 10 litres to stay happy and healthy. In smaller tanks filters are not really recommended because it would create a lot of movement in the water which bettas do not like however filters are essential in larger tanks such as 5 gallons or 20 litres and anything larger. Regular water changes are extremely important whether or not you have a filter.
No matter what size your tank it is essential for bettas to have a heater, they will not live long without one. Like all tropical fish bettas cannot tolerate fluctuations in water temperature, not only do the water fluctuations cause the fish extreme stress but also compromises their immune system putting them at risk of all sorts of horrible illnesses.
Your betta flaring at his reflection will not harm him, he may become a bit stressed at first but he will soon get use to his reflection and in time will probably ignore it.
Hopefully that is of some help to you, don't hesitate to ask should you need further advice