"That man is fat."
います and あります are used for talking about "existence", whereas 太い is an adjective used for describing something.
You can break down 太っています into 太って, the て-form (which is used to show consecutive actions) of 太る "to become fat" and います "to exist (for living things)". So the sentence actually kind of says "That man became fat and then exists (as he is now)".
On the other hand, 太い doesn't have that transitory idea. If you used 太いです, you're kind of saying "That man is, always has been, and always will be, fat", which isn't very nice nor necessarily very accurate. So it's not so much a method for distinguishing between living vs inanimate beings, it's just how Japanese people describe things accurately.
Yes, I can. A note up front: you're splitting the sentence in the wrong places, and the small っ indicates an extended consonant (i.e. it is not pronounced as 'tsu' here).
In what the Japanese call romaji (our alphabet) ふとっています is written as "futotte imasu". ふとって comes from futoru (太る), meaning "to gain weight/become fat", which in its 'te-form' usually indicates a state of being. This is complimented by います, the politer form of いる, "to be". So literally this means something like "being fat, [he] exists" --> "he is fat".
About probably "jin" problem. In this service, there are many international people. the words of country and people are frequently used. for example Spanish, English, Japanese. These words use "~~jin". So bot is not distinguish. hito is correct. jin is wrong. funny problem.
The hints that I get are ふとって and 太い (ふとい). In my opinion, only the second one is correct, since "fat" in English is strictly an adjective.
I understand why Duo has broken up ふとっています into ふとって = "fat" and います = "is", but I disagree with that approach as it fails to teach us that ふとって is actually the て-form of the verb 太る (ふとる), meaning "to grow fat, to become fat".