why not him?
Because the object of the sentence is the pronoun la, placed before the verb (l'), and the past participle changes to vista to agree with this. If it was him it would remain visto
Shouldn't it use proprio for my own?
I suppose that as long as Italian parts of the body are not referred to by using the possessives –as English does– to say miei in this case plays the part of own, being both expletives.
I saw it with my own eyes
Wouldn't that necessitate "proprio"?
In Italian, one's body parts are usually not refered to with possessive pronouns, so the existence of such is stressing that it is one's own.