I've taken both of my daughters from the age of 2, they're now 7 and 3.
Reckon on about 1mph and a maximum of 2 miles for a 2 year old. Olders ones can go a bit further but get bored easily so try and entertain them by taking them on walks with things to discover.
I'm a runner and a walker, I use Walsh's for running and some Merrel canvas boots for walking.
I agree with Baza stick to the right tool for the job. Fell shoes like Mudrocs and Walsh's are designed for running and don't have much of a heel or any support as this gets in the way of running.
Walking on the other hand especially long distances you'll need a higher heel and probably better ankle support.
Shin splints is caused by stress damaging the delicate tendons that bind the muscle tissue to the shin bone. This is typically caused by high impact forces from running.
Have you followed the golden rule and not increased your mileage (or time on your feet) by more than 10% per week ?
The other thing to look at is your technique and running shoes. Conventional wisdom (sponsored by Nike) is that you need thick soled 'cushioned' soles for stress injuries like shinsplints. More recent research, however, suggests that this is counter productive since fat soles and a high heel lift in particular impede the foots natural shock absorption mechanism which is to use the calf muscles as shock absorbers.
Try this free ebook for more information on correct technique and footwear.