Dready,
You need to be looking for a junior or expert size racebike, depending on your daughter's size- top tube lengths aren't an accurate way to size a bike, but if you aim for something in the 18 1/2"- 19 1/2" range you should be alright. (probably longer than your '86 Group 1!).
The usual brands offer decent completes- Redline, Kuwahara, Haro, Freeagent but it's probably best to have a look on the For sale section of BMXTalk where you might find something with a few upgrades or even a custom built race bike. The cheaper completes usually cut corners on stuff like bottom brackets, cranks and headsets so these will probably have been replaced on something that has been raced.
Gearing is a subject in itself, but for a 10yr old girl maybe aim for a roll-out of around 50ish, so 38/15 or similar. She should be on 1 1/8" tyres, 155-160mm cranks and handlebars not to high or you can't pull against them to get your power down- 5" rise or so.
Hope this helps!
Jamie