As far as I know, the metering circuitry used to be inside the batteries themselves, interacting with some corresponding device on the mobo. It just may be that these non-Dell batteries don't talk the right dialect, or are they Dell-approved?
The charging circuitry is on the mobo, yes. If it can be proven bad, it's unfortunately new mobo time. However, trying with a Dell battery may be a good step to take before this...
Measuring the power brick involves checking the output voltage against the nominal (noted on the brick, usually 18 - 20 VDC), but it has to be measured under load to be of any use and that might be a bit tricky if you're not used to that sort of thing. Basically, you need to connect a resistor of about 5-10 ohms (depending on the Current (A) or Power (W) rating of the brick) to the output and then measure the output voltage. Since the resistor will eat all the power that would otherwise go into the laptop, it'll become hot quickly. A 25 W type should last long enoug for the voltage measurement.
/RID