So, if you have a look at the tips related to this section (months/seasons/time), it becomes a bit clearer why this apparently random word shows up.
"Meán Fómhair and Deireadh Fómhair literally mean middle of the harvest and end of the harvest." This explains why the word "meán" shows up in this section.
"Listen - I've an idea" is fine to my ears. I think if the people building this course use contractions of the possessive meaning of "have" in this way, it's fair to assume that for them at least it's perfectly good English. People doing this course are spread right across the globe and their English-speaking ears are different. Some aren't native speakers, others are but in very different regions. I'm just happy if I can trust that the Irish is right.
If you use the "has got" construction, then "he's got" is common enough ("he's got the average", for example), but you're right, Duolingo has a problem with contracting "he has" to "he's" when there is no "got", and it's not just a problem with the Irish course - it crops up on other courses too.
I'm pretty sure that I sometimes say "he's a ..." - "he's a head on him like a jockey's ....", "he's a new job, and there's a lot of travel", "he's a new boat, so he's probably gone fishing". Even though I might say "he's" in these cases, I'd usually write "he has".
Of course, I'm not even sure whether "he's probably gone fishing" should be "he is probably gone fishing" or "he has probably gone fishing", but "he's asked me to help him" is definitely "he has".
The last example is using has as an auxiliary verb though. The earlier ones, using has as a possessive, sound totally weird to my English ears. I would always say "He's got a new boat/job or whatever". It's obviously a difference in idiom. Some commentators have objected that using 'got' in that way is ungrammatical and sounds weird.