Correct, but confusing and awkward. Natural would be Esteban’s cousin, ((his) aunt) Zumira’s son, …. The items in parentheses can be included or not, depending on how well the listener knows the family and whether he’s in it.

I think everything said so far is basically correct, but to sum it up, these would all be gramatically correct and logically equivalent: Esteban's aunt Zumira's son is going to be here. Esteban's aunt's son is going to be here. Esteban's cousin is going to be here. Esteban's cousin, his aunt Zumira's son, is going to be here.

And no - you don’t have to use the “s” on Aunt. John’s aunt Mary’s house isn’t far away.

I am a little confused by your question. I think the auto correct has played some tricks. But if you’re asking about two possessive cases in a row - yes, that is fine (and may be necessary). John’s aunt’s son is John’s cousin. The boy’s mom’s car wasn’t working, so they had to walk.