It's been a while since I've read into Korean accents, but if I'm not mistaken, some parts of Korea still pronounce ㅔ ㅐ differently. For the accent learner's are trying to achieve with this course, however, there shouldn't be a difference like you said :)
I think a lot of people are getting confused between 's = contraction of 'is' and 's = possessive apostrophe. In this Korean example, it is the latter ... that the food is for/belongs to the dog. E.g. 'my friend's food' does not suggest my friend is food, it suggests the food is meant for my friend or belongs to my friend.
N.B. the word 'is' is almost exclusively used to contract it is = it's, anyway.
If 의 is in a usual noun, like in 의사 which means a doctor, you should pronounce it as 'ui', rapidly speaking 'eu' and 'i'. But if 의 is used as a proposition, you can say it as 'e'. You should still write it as like '나의' or '학교의', but if you want, you can pronounce it like '나에' or '학교에'.
Here the <'s> does not intend to be a contraction for the verb 'to be', but it serves as a possessive. The food of the dog or the food for the dog.
Like in: This is my mom's house. It means the house is from my mom. Not that my mom is a house.
You need to understand it from the context or grammatical structure. Also I think in general Duolingo will write the full out. I don't think I've ever seen Duolingo contracting the to an <'s> before (of course I could be wrong in that).