Clarifications on a few points are in order.
EVENTUALLY you will be required to have the names on the adult tickets exactly match the name on the ID, but at this point many of the airlines don't even offer the ability to enter middle names or initials in their online booking systems. For that reason and for several others, it is not IMPERATIVE that the name on the ticket match the exact name on the ID for the forseeable future. TSA has indicated that they will be flexible as far as things like nicknames vs. full first names (Bob vs. Robert) and middle names/initials go.
Second, it's not the airlines that require proof of age for children traveling as lap babies, it's an FAA requirement and all airlines SHOULD be enforcing the requirement as there are penalties for non-compliance. That said, some airlines are better than others at checking consistently, but the best advice is that if you are traveling with a child that is an unticketed lap baby, you should be prepared to produce evidence in the form of at least a copy of a birth certificate that the child is under two years of age.
Other than that, all information about children under 18 not needing IDs is correct for domestic flights. Children will need some form of ID when traveling internationally.
