'He has yet more apples' is ok for me, but 'he's yet more apples' is wrong. You can only contract 'has' when it is being used as an auxiliary verb (with a perfective, e.g. 'he's got more apples' or 'he's sold his apples' etc). 'He has yet more apples' was actually the first answer I thought of when I saw the Polish phrase. I agree it is slightly archaic, but it isn't possible to contract the verb 'has' as this is the only verb in the sentence. You can do that with 'is' (e.g. 'he's tired'), but not 'has'.
This is a Duolingo-wide bug, I'm afraid. No one has put an answer "He's yet more apples" on the list of accepted answers. It is accepted because the algorithm knows that it's possible to shorten "he has" to "he's", but it doesn't know when it's possible and when it's not. So it just accepts it everywhere, and (what's a lot worse) sometimes even suggests it.
I didn't even know that, but apparently 'yet more' can mean 'even more'.
Anyway, with double meaning of "jeszcze", I wouldn't really understand this sentence your way. Maybe if it was "On jeszcze ma więcej jabłek", although that would sound a bit strange... I think your sentence would have to be "On wciąż ma więcej jabłek", using another word that means 'still'.
That's a bug. "He's" is accepted because "He has" can be shortened like that in "He has got = He's got". That's how Duolingo programmed it, and the algorithm doesn't distinguish contexts where it works from contexts where it doesn't :/ It's not an option that we put ourselves, so we cannot delete it.
You don't need to write three comments on the same thing. Also please read the other comments. The main answer is "He has even more apples". This is what the Polish sentence means. "She has a lot of apples. He has even more apples (than she does)."
Now, apparently "He has still more apples" is a correct sentence that means exactly that. And the Duolingo grading algorithm knows, that "he has" can sometimes be shortened to "he's". "He has overslept" = "He's overslept". But the algorithm doesn't understand when it's acceptable and when it's not, that's why it accepts it everywhere and sometimes even suggests it. Of course, "He's still more apples" is nonsense, but there's nothing we can do about it unless Duolingo developers fix this issue.
As for "He still has more apples"... for a long time, I couldn't understand why people try to answer like that. And then I saw a report saying "[...]My answer was "He still has more apples", using jeszcze as an adverb.[...]" - now I understand. Well, in theory you could treat "jeszcze" and "więcej" as two separate words here, although for 99% there are one phrase that says "even more". But okay, it is possible to interpret this as "He still has more apples, just like he had more apples last week". So OK, I will add your answer. But it really is an unlikely interpretation. For such a meaning, it should rather be "On wciąż ma więcej jabłek", which is unambiguous.