why η καρεκλα αυτη and not αυτη η καρεκλα?
Just another possible word order.
Both mean the same thing.
Does one mean more "this chair over here"?
No, actually not. It is the same
Why 'her own' is wrong?
Why both δική and της? Αren't they the same?
δικός, δική, δικό is similar to “own”. Combined with a possessive such as της “her”, you get “her own” — though often a possessive pronoun such as “hers” is appropriate.
Why not 'that' chair? 'This' is given in all the hints, 'that' isn't there at all.
I think that 'that' would be εκεινη, from what I'm learning
Correct, see here. The two are very clearly separate in Greek.