We've now cycled 27,000 miles together as a family - including a jaunt from Alaska to Argentina. Our twin sons are now 14; we started touring when they were 8.
The best preparation for a tour is as many saddle hours as you can fit in. That said, you'll get in shape on the road if you can't ride a lot beforehand. Just plan for short days when you start and you'll be fine. The physical fitness issue of touring is really a non-issue. I know that sounds odd, but biking is one of those things that you'll get in shape doing it. Don't let that one hold you back. Even your kids can do it if they want - it's way more mental than it is physical.
As far as equipment - nice equipment is nice, but not essential. Go with bikes that are comfortable for you. Going across the USA you will never be so far from a bike store that you can't hitch in if something does happen so don't worry about it. If you were going to go to outer Mongolia I'd say go with the best, but for what you're doing, don't worry. Just make sure everyone is comfortable and you're good to go.
About what the kids can handle - it depends on the kid. When we first took off when our boys were 8, we had them both on a bicycle built for three with their father as the captain. That worked beautifully.
The boys were 10 when we left Alaska for Argentina. At that point, one of our sons was ready for his own bike, so we put him on a single. The other boy was nowhere near ready so he rode a tandem with his father. It was perfect for both of them. My point is that kids are all so different that age has little to do with it - one twin rode his own bike through 15 countries; the other would have killed himself within a few miles if he tried it. He simply wasn't ready physically or emotionally to handle the stresses of it.
One of the things we did with Davy (on his own bike) was to sandwich him between us when we went through cities. My husband went first with the tandem so that Davy could watch what he was doing. I was behind Davy so that if he started doing something stupid I could yell at him. We only had one situation where we needed that - John could safely get over into the left lane so he jumped out, Davy followed but the car was too close. Cars started honking their horns and I started screaming and Davy jumped back over to the right. We talked with him a lot about how he - and he alone - is in the that particular place in time and space and he needs to always check for himself rather than blindly following Daddy. After that we didn't have any more problems.
The speed issue won't be much of an issue. People are inherently cautious and will hit the brakes when they are going too fast. As you gain skill and get to know exactly what your bike can handle, you might speed up but it's downright scary to go too fast and those brakes are right there. If you're concerned about the kids not knowing what they can handle, mandate that they stay behind you and then you can set the speed.
I think you're bang-on on the timeline. 3 - 4 months should give you enough time to get across the USA. It really depends on how much you want to pedal. We've found we've slowed down a LOT since we first started touring. We now do around 500 - 600 miles/month rather than the thousand we used to do. That gives us more time to play, which is what we want.
Good luck with planning! It'll be a BLAST! Feel free to send me an email to familyonbikes at gmail dot com if you want. We've got tons of info at our website (familyonbikes.org) as well.