At half power what should I expect to read for Battery voltage?

3.5V is a very low voltage -- that's the minimum cutoff for the "low battery warning"; any lower than that and the device won't even boot (it'll just show the red empty battery symbol while it charges above 3.5V).

Charging voltage between red and black should be 4.1-4.2 volts. The white wire is not a reference voltage -- it's connected to a thermistor that cuts off charging if the battery overheats. If it's not connected properly, then the battery won't charge; I suspect that's what's happening.

(I can't give you a voltage that means "50% charge" -- LiPoly/LiIon batteries measure around 3.7V for most of their normal service range. Check out how flat the green line is on the first chart at http://www.mpoweruk.com/performance.htm !)