たかい and ひくい are いadjectives. This means that they will always end with です or ~くありません.
Example: たかい です。たかく ありません。
However, ふとる(ふとっています)is a verb. When you use the te+imasu form of a verb, you are describing a current state. There is no te+desu form. It simply doesn't exist.
I think the reason it is confusing is because "fat" and "tall" are both adjectives in English. In Japanese, on is a verb, so it will behave differently.
Now, on a more general note. The difference between です and います・あります is actually pretty simple.
です identifies something. ペンです=Hey, this thing right here, this is called a "pen". (Hence why we use it to introduce ourselves.)
います・あります identifies the existence of something. ペンがあります=Hey look, there's a pen right over there.