Yes the 24-120mm is an FX lens and will work fine, but given the timing you mention it's obviously an earlier model than the current f4 VR version. The older versions are often reviewed as average performers so keep that in mind. The range is nice and generous and I'm sure your daughter will take some fine pictures with it but it's not an outstanding lens.
The newer version on the other hand is much better.
As for putting a DX 18-200 on a D600, well yes it can be done. You can shoot the D600 in DX mode and get 10MP cropped files. But this will only be satisfactory for a short while. You'd be using the D600 at half capacity, like buying a Ferrari and putting diluted ethanol in the tank. Sure it'll get you from A to B but not in the style a fill up of high octane FX glass will!
Ideally the 18-200 should stay with the D40 and the 24-120mm with the D600. If you're out one FX lens now and looking for an inexpensive add-on why not get a 50mm f1.8? That would be a great addition to the D600 without breaking the bank, and there's many other possibilities like a used 35-70mm f2.8 D and so on...