In nominative form:
That = εκείνος, εκείνη, εκείνο (sing.) and εκείνοι, εκείνες, εκείνα (plur.).
This = αυτός, αυτή, αυτό (sing.) and αυτοί, αυτές, αυτά (plur.).
So ,by contrast, 'THIS child eats' would be 'αυτό το παιδί τρώει'??
No, το παιδί is the child, the children is τα παιδιά.
I am not sure about εκείνο but there is εκεί
No, because: 'There the children eat' would be: " Τα παιδιά τρώνε εκεί" :)
Εκείνο sounds exactly "equino" in Spanish. Is equine ιπποειδών in Greek?
Yes, it is! ;) Ιπποειδών is the genitive case
Could it also be: 'There the children eat' ?
Why not "the child is eating?"
Because "εκείνο" means "that", not "the".
I don't really undersand why εκείνο is used. What would be the difference with αυτό?
"Εκείνο" is used about things that are further away from the speaker compared to "αυτό". Also read some older comments between me and Walt1965 in this discussion.