if you going to use filter for your 3 gall you will need to do 50% weekly water changes , i prefer more water changes though 50% x2 week with gravel vacuuming. You also will need to swish, wash filter regularly.
If you decide not to use filter you can do 1-50 and 1-100% water changes for your tank. Which i think is easier because you don't have to vacumm the gravel and take care of the filter media. I personally think it easier and better. And be careful with filter if you using it, make sure it baffled. I saw a lot of people wrote that betta got sucked in the tube.
If you think that he ripped his tail i would take cave out.
I don't like when betta hides a lot. Make sure he don't hide a lot or stay at the bottom of the tank, or under the plant, then something is bothering him. They do it of course but not all day long. Usually betta is pretty active fish.
I don't know my personal opinion he is new and you new to betta i would turn the filter off and just watch him make sure he is eating and active and healthy, and not hiding. Make sure he is warm. But if you just bought your heater i would check on it. I had heaters that malfunctioned. I always check new heaters before i put betta in...