It seems to me that you are doing a pretty good job taking care of your betta. I might sometimes do a 35% water change, though. I typically do around 35% weekly for my 5 gallon.
According to the information, your betta seems to be ripping his tail. This could be on any decor, I recommend silk or live plants, or other decoration. It needs to be soft and no sharp edges. I recommend feeling the decor and removing the sharp ones if needed.
If this is not the case, your betta is probably biting his or her tail. Some long-finned bettas may bite their tail. This could be happening because:
*They are just upset their tail is weighing them down making it hard for them to swim
*They are stressed
Your betta biting it's tail may be a lifelong habit. I actually have a betta who does the exact same thing often. Or it could just be stress of a new environment. The best chance to try to get your betta to stop tail biting is:
*Give your betta places to rest so they won't get too tired swimming with a long tail
*Reduce the filter strength (Maybe put a sponge over it?)
*Reduce stress overall
Normally, bettas biting their tail but still acting like a normal fish doesn't really impact the fish. As long as you keep up on your water changes - and possibly change to 35% weekly - your fish will do fine. A bitten tail or ripped one is easier to get infected, because it is more exposed so water changes are important.
I wish you the best luck with your betta, hopefully it lets it's long, beautiful tail grow back!