Bus is not an option unless you want to spend hours in the heat with your kids, with an hour or two more standing/sitting inside the bus till you get to your destination.

Taxi, MAY be an option BUT be prepared to wait forever to hail one. Even if you call one and its peak hours, you won't get one to come to your door.

Yes there are female taxi drivers upon request (you gotta call one)
Car seats? no-so-sure

Walking in Dubai is alright DEPENDING on where you live. If you live downtown or somewhere close, you gotta walk in knee deep sand sometimes or on the street cause there is a whole bunch of construction.

Best advice is to hire a driver with a car to come and pick you up when you call and wait on you.