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?
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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.