I agree with the advice given. I think his illness is due to stress, from being in a small bowl. Loss of color is a symptom of stress. Consider putting him in a 5 gallon tank with a heater and a sponge filter. This can be set up easily and cheaply.
For now, I would treat in the bowl. If you can find a container that's bigger than one gallon, that would make medicine dosing easier. It doesn't have to be a fishtank. It can be a Rubbermaid container. (but I wouldn't put a heater in a plastic container!)
Keep the temperature stable. If there is no heater, the temp rises and falls when the lights are turned on or off. That may only be a few degrees, but it's a lot for a little fish. If you have to keep him under a light day and night to keep the temp stable, go ahead. Put a washrag over the top somewhat to give him some "shade".
Test kits are a must but for now the
lfs can do the testing for you, if you take a water sample in. If the ammonia and nitrite are not high (above .25) try doing smaller water changes. Large water changes can be stressful. In a small bowl especially, the water motion can cause stress during a change. Do maybe a few cups a day, instead of 50% or more changes every few days.
Bettas also must be able to get to the surface to breathe with the labyrinth organ. They do use their gills like other fish, but they must exchange air at the surface also. Since he's in the bowl, he shouldn't have a hard time getting to the surface. I lowered my tank water level to help my betta get to the surface once, and he perked up. Hopefully by keeping the temp stable and not stressing him out during a water change, he will perk up and come to the surface instead of lying on the bottom.
I wouldn't worry too much about dropping him. When I got my previous betta, the clerk dropped him from about 5 feet (he was in a high tank). He was fine and didn't show any problems from it. Another betta went sliding into the sink headfirst! I really don't think it's the cause of your betta's problems now.
I would also recommend an antibiotic, since his gills seem inflammed -- the rapid, heavy breathing could be a sign of this. I have had good luck with Kanacyn and Jungle Fungus Eliminator. Kanacyn is a capsule (one capsule per 10 gallons) and you'll have to open the capsule and measure out enough powder for the number of gallons your betta is in. It's kind of tedious, but that's a good medicine. The Jungle product is also good. It is a yellow, loose powder that is easy to measure. If you can find one of these meds, I would use it instead of the tetracycline. Use it for 5 to 7 days, following the label directions on the frequency of doses, and changing a few cups of water before you medicate.
Good luck!