He's = he has, but only when "has" is an auxiliary verb and not when it's a main verb. (apart from in dialects like in Irish English). Duo can't tell these two contexts apart - it happens in the other language courses as well when Duo generates "correct" or "alternative" answers.
I believe that in most parts of the English-speaking world "He has a great teacher" would not be shortened to "He's a great teacher" partly because most people would understand the latter as "He is a great teacher". But I suppose there are areas in the world (UK?) where this is OK. Otherwise shortening "he has" to "he's" when followed by a past participle as in "he's been a great teacher" is normal. Feel free to correct me if I'm wrong :)
Not really. Personally, he's tends to mean he is (i'm from uk lol). As mentioned, the only exceptions to the rule are predominantly past tense contractions. S/ He's eaten, sang, slept, washed, rode, written, drank etc... If you followed he's by "got" it would make more sense. He's got great teachers etc...
A "tapped r" is made by touching the tongue at the same point that is used for "d" - the contact is just less firm. The two sounds are very similar. If context leads you to expect a "tapped r", you'll hear "r", but without context or without previously knowing the word, it's totally reasonable to interpret the sound as a "d".