Great philosophy. I can tell you from personal experience that 'taking better pictures of your child' is a justification that usually has a lot of traction with a wife, at least with mine.
That's a great idea. Some introductory books can be great as well - Bryan Peterson's Understanding Exposure and some of Michael Freeman's books are great places to start.
Definitely a good investment, but perhaps not something you need right away. My experience with the 70-200mm f/2.8L IS II was that on an APS-C camera, the focal length was rather 'awkward' - it was too long to use conveniently indoors except right near the wide end. As a result, I really didn't use it that much on my 7D. After getting a 5DII, though, it became my second most-used lens - on FF, it's a great focal range indoors and out, living room to backyard to beach.