82:9 We may place food before a non-Jew in a courtyard or in a house, even though we know he will take it out, so long as we don't actually place it in his hand. If one puts the food into his hand, it's like performing the "uprooting" portion of transporting. This applies when the non-Jew has permission to eat the food there, should he so desire. If he doesn't have such permission, or if he's given so much food that he can't finish it there and must take some "to go," it is prohibited because it appears that he was given the food specifically so that he would transport it on Shabbos. This is equally true for any other items that the non-Jew would clearly take outside.
82:10 A woman may support her child walking even in a public domain so long as she doesn't drag him. Rather, the child should raise one leg while the other rests on the ground. He should stand on one leg until the other is back on the ground so that he is always supporting himself with at least one leg. When his mother drags both legs, this is considered carrying the child and is prohibited even in a carmelis (the quasi-public domain). One may not carry a child, even one who is big enough to walk on his own, not even in a carmelis. Yes, there is a concept of "chai nosei es atzmo," that a living thing carries itself, but the context of this statement is only to exempt a person who did it from having to bring a sin offering, not to permit one to do so at the outset. Carrying a child is a rabbinic safeguard, as is carrying in a carmelis, so people may mistakenly be lenient in this matter. They should be informed otherwise.