[этот стул] is one construction. Both parts of it have the same function in the sentence. [это] (есть) [стул] is two different phrases that have two different functions in the sentence (subject and predicative) separated by the verb. The tricky part is that there's usually no verb in these kinds of sentences, so they can be hard to recognize.
Have you seen the guide to loving and liking by olimo: https://www.duolingo.com/comment/11754722 ?
If you see that chair often, and you don't like sitting on it every time, you can use both. However, if you see it for the first time and don't like it 'at first sight', then you can only use «нравиться».
No, that would be «табуре́тка» or (more formally) «табуре́т».
«Стул» can only be translated as «stool» in medical sense.
Well, that's just because both come from the French tabouret. Since French was the lingua franca of the upper classes in the past centuries and France was the leading country in matters of fashion, furniture, etc. it is not surprising that many words in languages all over Europe and the whole world have a common French origin.
I'm just speaking for myself here, but I can mostly tell when to use a different form. Some (like у меня нет + gen) aren't as obvious. But you can find convenient lists online:
I'm not sure how far you are in German, but German does this too. Like with helfen or Geben (help and give, respectively). We actually do something similar in English, we just don't think about it.
Example: I am giving HIM a present. (Not I give he a present).
DEUTSCH: Ich gebe IHM ein Geschenk. РУССКИЙ: я даю ему подарок.
You'll get the hang of it. Russian is quite different from English. It just takes time to get used to it. :)