I just looked up what Betta bowl essentials is. The instructions says 5ml per quart, so for a gallon, you would need 20ml, I suggest you add more and refill the right amount everytime you do a water change.
The tear in his fins could be from the plastic plants, I suggest getting silk plants.
I'm not sure what the black spots are, can you upload pictures?
Bettas have special lungs that can take in oxygen from the air and water, so you will see them breathe at the surface.
New bettas tend to not eat the first few days since they are adjusting to the new environment, they can go a few days without food.
I suggest getting a slightly bigger tank unless you can keep up with the water changes. I think for 1 gallon, it's 2 or 3 100% changes a week