gift on shabbat

Question

a)may one purchase a gift for a specific person and give it to him/her on shabbat? Am I correct in thinking it is allowed, because when he purchased the item he purchased it for that person and therefore was "zoche ladam shelo b'phanav" from before shabbat and is not now making a kinyan on shabbat but merely handing over the item he already designated for the recipient before shabbat. b) would the halacha be different if you were giving it to a child? c)what if you just bought it randomly and then on shabbat decided to give it as a gift to someone. Is it then asur to give it to them? how about a child in this case? by the way for the person asking about why the o-u puts a dairy where it doesn't belong, they even admit on their website that there policy is contrary to halacha. A quote from the o-u website ("According to halachah, a dairy component which is less than one part in sixty may be botel nullified. Nonetheless, as a matter of policy, the OU will not imply a product is pareve by printing an OU without a “D” on the label, even though the dairy component is at trace level.") - quite humerous.