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Regarding the law that one may carry a child who can walk by himself outside without an Eruv, how well does the child need to be able to walk? Let’s say he walks only with assistance.
Actually your question is based on a common misconception – that a child who can walk by himself may be carried outside on Shabbat. In truth, the Talmud (94a) has a notion that a living creature “carries itself” – i.e., assists the carrier slightly while being held by holding himself in such a way as to make the carrying easier. But this only means he is helping the carrier – not that he is literally carrying himself. It is rather akin to two people performing Shabbat labor together. And when two people do labor together, they still transgress Shabbat, although only Rabbinically. (Many more details to this principle, and in fact some understand the leniency that a person carries himself entirely differently.)
Thus, we may not carry a child outside on Shabbat where there is no Eruv. You could thus only take your child outside if he will actually walk by himself. You can hold his hand and nudge him forward if necessary, but you must take care that you are only helping him walk, not actually moving or lifting him.
(Sources: Shulchan Aruch O.C. 308:41, Mishna Berurah 154.)