...and giving H something to do rather than just handling it myself, in an effort to make him feel "needed."
I'm going out on a limb here to say that more than feeling needed, that he wants to feel love and that he is contributing in a way you appreciate.
Please let him know when he does something from the list that you love him and appreciate him doing that. Whether it's off the list or anything in general.
Praise the positive rather than punishing the negative.
