The 2-gallon tank sounds great! That would be a great improvement to what most betta fish live in. For now, you don't need a filter, and if you have some extra money then you could buy one, but for a betta, I don't think it is nessisary
Look at the ingredients on your food. Get back to me once you have that or, it will be something like shrimp and you will not need to. Or, it could be a digestive issue.
The chances of this happening are so small! the only reason a tank would break is if it was somehow damaged. If you wanted to you could reseal the outside if you are really worried about it
You can do plants, and I would say around 120 small fish (1inch and below) or 7ish larger fish. I don't know what kind of fish you want, but any extra preference info would help!