Everyone I talk to says you should lime your lawn twice a year, once in the spring and once in the fall. But according to my research, you not only do not have to lime every year (as you say, a ph test is best to determine soil acidity), but you can lime at any time, except for the 2-3 weeks before or after you've fertilized. (I wonder if it's the lime industry fueling these myths? ;-)
That being said, here in Halifax Regional we are not permitted to use the weed & feed products due to the by-law, so we're adopting natural control. That means pulling dandelions by hand (the nifty Weed Hound tool makes that easy) and practicing good turf care: watering, fertilizing, overseeding, aeration in the late fall, and just generally trying to keep things robust and healthy.
We have weeds, too, but as long as they are green I don't pay them much mind. Every now and then I'll go out and dig some of them up, spread some top soil and grass seed, cover it over with a layer of peat moss and then soak it with the hose.
If you're worried about the kids, I'd wait to spread the weed n' feed. I'm no expert on the toxicity of those products, but perhaps it's better safe than sorry.
This site has some good tips for natural lawn care: