A piece of tail that big is from being caught on something and ripped off, possibly by the rock in the tank. Aquariums with long finned fish should not have plastic plants, rocks with jagged edges or anything that can catch the fins and rip them.
The red and blue betta does appear to have white spot. You need to raise the water temperature to 30C to kill the parasites. If the water is 28C, the parasites do not die and will simply reproduce faster and kill the fish sooner. So raise the temperature to 30C and keep it at 30C for 2 weeks.
Depending on where you buy the water from, it could have anything in it including chlorine. If you are going to buy water, buy reverse osmosis or distilled water so you know it's free of chemicals, including chlorine/ chloramine.
If you contact your water supplier for your home, you can ask them what the pH, GH
(general hardness) and KH
(carbonate hardness) is. You can also ask them whether they add chlorine or chloramine to the water. You can look on their website for this information (Water Analysis Report) and post a copy of the report here for us to go through. If your tap water is suitable, it is usually cheaper to use a dechlorinator with tap water rather than buying water every week.