I have some mixed emotions on this one
...my primary concern would be the children..Is it an affordability thing or a green (enviromental) reason for not having a vehicle. My only thing is you'll subject your children to some weird weather (as you stated) and what if there is an emergency and the children need to be rushed somewhere...you can't really package them up and rush them to the hospital.
I know absolutely nothing about living car free other than when i was younger and poor.
Hope it works out for ya, best of luck bro.