Bettas like small tanks. My betta did really well in a 2L glass water jug. While they do like smaller areas, if you wanted to move him to a bigger tank I'd still make sure that he has enough hiding spaces to feel secure. I usually don't worry about problems with Ich, if bettas are cold they'll develop fungus first (I don't want to scare you, you'll know it if he has it, he'll be fluffy). Major color changes are a concern, but slight ones are normal.
The other thing I was wondering is, how often are you feeding him? If he's in a small container and you're feeding him twice a day that's too much, and the water conditions could affect how he's feeling. Size of eye = Size of stomach. Literally.
This is NOT true at all. Where did you get this information? Bettas, like any other tropical, need space to thrive and don't live as long in smaller tanks. I have seen bettas kept in little vases/bowls and noticed that their coloring is more dull, fins are often clamped, and they tend to be listless. Of course, you can't notice a difference until you keep them in a larger aquarium.
2L is too small for ANY fish. Ammonia will build up really fast in that size container, stressing out and often killing the fish (even if it takes a while). You would literally be having to change the water a couple times a day to keep ammonia down. But that would be so stressful that it wouldn't work anyways.
How would you heat a small jug safely? That's a big problem. You can't. They don't design good aquarium heaters for such small amounts of water.
Also, it is not true that if they get cold they'll develop fungus. True fungal infections are actually pretty uncommon, what most people think is fungus is really a bacterial infection (which can be caused by many factors, one of the most common being poor water conditions). A fish can easily get Ich in cooler temperatures and it would actually be more dangerous since the parasite's life cycle will be lengthened. Ich is usually brought on by stress and cold temperatures are stressful.
I'm sorry, I just don't think it's right to recommend keeping any animal in a small jug. It's equivalent to keeping a puppy in a hamster cage.