All well water is different and some can contain heavy metals. You can age it and it will remove chlorine ans stabilized the ph but the only way to get rid of the heavy metal is the water conditioner. So you will need to use water conditioner.
I don't know a lot about well water , and i am not sure if Hallyx knows , so i did ask Sakura8 to look at your thread and give you more information about well water. I think you can use it without problem with the water conditioner. I am not sure if you still have to go to the pet store to check it though, but if you go to the store any way why not. Just take a sample without water conditioner in it and check it.
Again when you transfer him from the water that you are using to the tap water do a few (5-6) small portions water changes about every 30-40 min to make him get used to the new water.
Your betta looks fine, i don't see any problem . I think the color that you was talking under the gills and on the stomach its his normal coloration . I think a lot of bettas his color has it. So its not a velvet that i mentioned before.
The second picture of the heater is fine. It for up to 5 gall tanks. The only thing it will not of course fit in your current tank, but it good for 2-5 gall tank
The temperature in your house -you don't have to have 80* 76* is fine for him.
With new heater you need to test it in a bucked of the water for at least 24 hrs before you put it in the betta tank. Let's you find out if it will hold a decent temperature , or if it defective. I saw a few post when new heater leach the fluid in the tank and harm the betta. So yes it good idea to test it. You know what you can do . If you planing to buy bigger tank you can just check new heater in that tank. If you not going to buy new tank then you will need smaller heater because this one not going to fit in your current tank