could I say δεν αρεσω το κοτοπουλο as an alternative and if so what is the difference between the two alternatives?
I don't think so, the reason being that, "δεν μου αρέσει ..." means that it doesn't please me. I think the μου is required.
Actually the meaning here is that the chicken is not pleasing to me. "To me" is translated into the indirect object "μου". The structure here is a lot different from the english one but by time you get used to it.
don't is slang in English do not , is not . However should you point out I did not use " I " , there is no question mark in either response , so seems to me my answer is acceptable .
Sorry, to be so late in responding.
"don't" is simply a contraction and exactly proper English, it's not slang at all.
As for why your sentences was not accepted please note that we the moderators have no way of seeing what the exercise was, what you wrote nor what the response from Duolingo was. Without that information we're unable to explain why something was not accepted.