I have two children, 8 months and 3 years, and have been car-free since before the first was born. We have the advantage of living somewhere very well served by public transit (which is why we moved to this neighborhood) but the disadvantage of living in a city where motorists are fairly hostile to cyclists. We use our trailer (a Chariot Cougar, I looooooooooove it) extensively for kid and grocery transport, and plan when the older child is big enough to get a trailalong and a bike seat for the younger.

For your wife, I would strongly recommend looking at what she's wearing to cycle in and how fast you're riding. I commute to work in office-appropriate clothes, but choose light-weight, fast-drying synthetics. On really hot days I just slow it down and carry a hankie for my sweaty face. At my destination, I use the little rice-paper facial blotters made by a few companies (like Burt's Bees) that mop up sweat. I find I'm more sweaty and uncomfortable wearing what many casual cyclists think of appropriate, such as cotton t-shirts.

Like someone upthread said, our biggest problem without a car is getting to recreational opportunities. We use rental cars or carsharing for that, since even destinations served by commuter rail wouldn't be possible for us if we wanted to bring the trailer.