You can find ponies for the ride from tatoo to fairy meadows. Sorry, no idea about price. And you probably would need to hire atleast one because the two smaller kids wont be able to do the trek, unless you carry them.
The trek from fairy meadows to beyal is relatively easy one and shorter (around 1.5 hrs). You probably wont need ponies for this part. Going to base camp with kids is a bad idea. Maybe a few of you can go while others stay at beyal.
I dont want to scare you, but some of the trek involves a very thin walkway right next to a thousand feet drop on one side and a cliff on the other side. They are only short strips of such walkways so dont get afraid, unless you have extreme fear of heights. Also the start of the trek is rather steep. Definitely not a place to go with little kids with you. I think you can delay that decision to when you reach there and what physical condition you are in.
There is no electricity there. No medical facilities at all. Fairy meadows is nothing like murree or ayubia...its wilderness, mountainous, and rather steep trek with only summer settlements of a few nomadic families.
One big advice you must follow religiously....only take the material you absolutely need. Remember carrying weight on a trek makes it much more difficult...unless you intend to hire porters, you must have only the cloths you are wearing with additional jacket/themal leggings and some food and water (ofcouse plus things for the kids).
Secondly, hire wooden huts instead of tents as they are much warmer particularly for the smaller kids. It can reach as low as 0C at night in beyal with chilly winds since beyal forms a valley right in front of Nanga Parbat. On the positive side, it offers breathtaking views of the mountain. As far as the sticks are concerned, they aren't a must though helpful. Some people even consider it a nuisance you have to carry. Do, however, keep extra batteries for you camera (rechargeable wont help since there is no electricity).