Then how does one say: "This horse is mine"?
"Αυτό το άλογο είναι δικό μου"?
What is the difference between το άλογο and ο ίππος? the latter seems to be more ancient (cf. English and international words like hippology, hippodrome, hippogriff) but I might be mistaken.
Does "mou" change in conformity with cases or it's a noun that changes, or an article? Or everything?
Μου is the weak genitive form of the first person singular pronoun εγώ and is used to indicate possession. It does not change under any circumstance, if what you want to express is possession.
So only articles and nouns change in declensions of possessive constructions?
Yes for example:
Ο άντρας μου=my husband. (Nominative)
Του άντρα μου=my husband's.
Τον άντρα μου=my husband (Accusative)
- άντρα μου= (vocative) my husband
Is there a version shorter than 9 syllables, or is Greek usually more long-winded than English? ("It's my horse" is 3.)
Greek is always more long-winded that English, and that's why it is spoken fast.
I put "the horse is mine". Why is that not quite correct? Please and thanks!
It doesn't express the same idea. In the Duo sentence, it refers to "this horse" in other words a specific one. It is a slight difference but in a teaching course there needs to be accuracy.