OK, so is this a good summary of what's going on?
- swimming/buoyancy problems
- a small patch of grey on his left gill that you described as "peach fuzz"
- 10 liter filtered tank (about 2.6 gal)
- about 21°C (about 70F)
- water changes every three weeks or so
Here are some suggestions:
Take a deep breath.
Don't panic. We'll get this sorted out. It doesn't sound like he has anything major going on.
Increase your water changes.
A 2.6 gal tank with a filter should have a 50% water change every week. (NOT every 3 weeks or so.)
Over time, toxins, wastes, ammonia and bacteria build up in the tank. Your filter takes care of some of this. But the only way to remove the rest is via frequent water changes. So I would do a 50% water change every week.
Clean, warm water is the best way to keep them healthy.
Add a heater.
Bettas are tropical fish. As redthebetta pointed out, they need temperatures in the range of about 24-29 C (76-84F).
This is the most significant issue he currently has. White fuzz can indicate a bacterial issue called Columnaris. The easiest way to treat this is with frequent water changes. Changing the water will remove the bacteria that causes Columnaris.
Your low water temperature is actually working in your favor for this. The cool water slows down the rate at which bacteria multiply.
I would still try to raise the temp a little, so that he's more comfortable. But I'd try to bring it to about 24C (76F) right now.
If he didn't have this, I would recommend a small amount of Aquarium salt to treat the Columnaris.
So let's try this alternative:
1) Do DAILY partial water changes. Remove about 50% of the water every day, and replace it with fresh (conditioned) water.
2) Add a small amount of Epsom salt to the water.
3) Keep the water temperature about 24C.
Partial water changes will remove the Columnaris bacteria from the water. And the 24C temperature will keep it from multiplying quickly.
Epsom salt is available at supermarkets and pharmacies. (It's not sold at petstores.) Look for 100% magnesium sulfate. (No dyes, additives or perfumes.) It's available at Walmart, CVS, Walgreen, Publix, etc.
Here's how to use the Epsom salt:
- Fill a container with one gallon of water that is at the same temperature as his tank.
- Add water conditioner.
- Stir in 1 teaspoon of Epsom salt. Make sure the salt is completely dissolved.
- Remove 50% of the water from his tank. (This is about 5L)
- Slowly, over the course of about 30 minutes, pour the Epsom salt solution into the tank. (Taking longer than 30 minutes is fine.)
Repeat the above process every day until both the "peach fuzz" and swimming/buoyancy issues are resolved. BUT don't use Epsom salt for longer than about 10 days.
IF AT ANY TIME, the 'peach fuzz' becomes worse, or his swimming problems worsen, let us know! He may need additional treatment if this occurs.