The bulge near his tail is his swim bladder. It looks OK to me.
IMO, a 1 gal tank is OK, if you do frequent water changes. Changing out the water frequently is the only way to remove toxins, ammonia, wastes and bacteria that build up.
Do you know the temperature in the tank? Bettas are tropical fish, and need water in the range of 76-84F. If the water temperature is much lower than this, it will slow down their metabolism, and make them susceptible to illness.
If you don't have a thermometer in the tank, I recommend getting a "floating glass" one. These are available at Walmart, Petsmart, Petco, etc. They cost about $2 to $3.
Finally, the food that you feed is important. A betta's natural diet consists of insects, so they need a lot of protein.
Look at the label on your food container. If the ingredients contain a lot of grains/fillers (such as wheat, corn, soybean, gluten, potato, meal, etc), consider switching to a higher-protein pellet. A diet high in grains/fillers can cause constipation, bloating and buoyancy issues in some bettas.
On this forum, the two most often recommended foods are New Life Spectrum betta pellets and Omega One Betta Buffet.
These provide good nutrition for them.