Virtually all colors look dated after a while, although neutrals like gray, taupe, beige, white are pretty much timeless. Just go with what you like. It would help to have some idea of your furnishings, however. And remember - like all colors - there is a wide spectrum of 'gray' out there.
Re the babies' room, white with a lime green accent wall could look great - or it could look awful. Depends on the paint job and how well you coordinate the furnishings. That combo is not 'girly'. You could pair it with pink or lilac and make it girly, however. Again, go with what you like; the kids are too young to care. In any case, they'll probably want a change in a few years.
Like most teenage boys, your son doesn't care about paint colors. He said 'blue' just to get you off his back. I can see his eyeballs rolling from here.

Boys tend to like navy blue. Personally, I would do neutral taupe walls with navy blue and white bedding, maybe with red accents.