"그는 자기의 집이 있어요."
Translation:He has his own house.
That is correct. Save with "have"--"You've been to Korea" or "They've been to Korea" are acceptable, while "I've a house" or "They've cars" is not. (I think I've seen "have" contracted outside of Present Perfect in some archaic texts, but even in archaic texts it's very rare.) However, "has" is even more unacceptable, because the contracted form "s/he's" is also the contracted form of "s/he is" and would be read that way in any usage other than a Present Perfect. So "He's his own house" would be read as "He is his own house"--I didn't even realize the contraction was intended to be "he has" at first. I think "had" can only be contracted in Past Perfect. So, "I gave him a DVD, but he'd already seen it" is acceptable, but "He'd a car until he sold it" is not.
For these lessons, when a person is implied, 그 can be assumed to refer to “he” or “him.” 그녀 is its feminine counterpart.
Assigning gender to pronouns is quite new in Korean though (same in Chinese, which did not have a separate character for “she”/“her” in the century prior). And if my small group of Korean American friends are representative of the whole Korean-speaking population in general, they say 그 is gender-neutral.