If a father must carry the sins & punishments of his son (under bar mitzva) as we know he does from the brocha of "Baruch Shepatrani Meonshin shel zeh" does he — and if not, why doesn't he — "carry" his sons mitzvos and get schar (reward). If he is fully responsible for his son in one factor, why not in everything else?
Actually, the father (and mother) do carry the Mitzvos; they get rewarded for their children's good deeds. Not only before but after their children's bar/Bat Mitzva as well.
And that is the reason for orphans saying Kaddish; every Mitzva they do is credited to their parents. The Kitzur Shulchan Aruch recommends that a person's will should include the request for his kids to "shine" in a specific Mitzva, so that he can be credited for years of that Mitzva being done.
For that matter, if you're responsible for any person doing a Mitzva then you get some reward for that Mitzva.
On the flip side, if you're responsible for any person doing a sin then you get some punishment for that sin. If so, why the "Baruch Shepatrani Meonshin shel zeh"? Because until Bar Mitzva, a father gets punished for not educating his children; after bar Mitzva it becomes the child's responsibility to educate himself, and he can no longer blame his sins on the deficient education his parents gave him.