Tyres: go to Schwalbe's US site and trust their descriptions of tyres absolutely. For they are German and engineers:
http://www.schwalbetires.com/bike_tires/cyclocross
http://www.schwalbetires.com/bike_tires/road_tires
The tech people at Schwalbe told me that the best bet in their line for dual use is the Extreme and I use it myself. It's got excellent road handling - perhaps the safest road tyre I have ever ridden because of the wet grip, although definitely not the fastest - and is good on any sort of dirt other than real mud, but it is very expensive (£70-£80 a pair!) and won't be quite as fast offroad as real cross tyres like the Rocket Ron. The Marathon Cross might be a good cheaper alternative.
Using real cross tyres on the road will get expensive because they'll burn out faster than the dual use tyres (ratings like "durability" only apply inside a category - a 4 star durability cross racing tyre won't be as durable as an Extreme.)
Anyway: the right tyre choice is critical for road safety and racing speed. This is probably your trickiest area. I don't have a daughter but I do have two cats that I'm very fond of, and I wouldn't let either of the cycle regularly in London traffic on, say, Michelin Muds. If you do go for a real cross tyre, make sure it has at least a 4 star rating for "hardpack" under Grip - and even then, test it yourself on emergency braking and cornering.
Re the linked bike - it's a 24 inch wheeled bike for 7 to 11 year olds. If this is the right size for your daughter there is no way she can ride a 700c frame. Or vv. Also: it's a road bike and probably won't have the ability to run wide cross tyres - if you could find them in 24''.
About second hand parts - you almost certainly know this - don't buy second hand stems and bars. Or if you do, be very cautious. You don't know what stresses have been put on them, and alu stems and bars can snap in use; I'd only ever use a used stem to try out a particular size and angle briefly and very cautiously. About the only things that you can buy second hand and save much money on are the bar cons, it seems to me. And a spare set of wheels if you go that way.