How to measure voltage differential across different altitudes
Q1: I imagine you could measure the potential difference using weather balloons. Send one balloon up to 10,000 ft (Balloon A) and one to 20,000 ft (Balloon B) with wires connecting them to a voltmeter. The challenge here is to account for the wire resistance, which, if the potential difference is low, will in all likelihood interfere with the measurement. If the difference is too low to be measured, you would have to amplify the voltage and then account for the amplification and wire resistance.
Q2: I would try using weather balloons again, but obviously Balloon A and Balloon B would be at the same altitude but 10 miles apart from each other.
Q3: Instead of a voltmeter measure with an oscilloscope. There's going to be noise no matter what; what oscillations are produced by noise and what is actual oscillations is a judgment call. I would say that truly oscillating potential difference is likely to be low frequency, so to filter out high frequency noise connect the circuit to a low-pass filter before processing with the oscilloscope.
Another way to determine oscillations is to measure the voltage with a data logger and post-process with computer software; any programming language that can easily manipulate large amounts of data would be sufficient, even excel would work. Plot the voltage vs time using a linear regression model to smooth the noise out. As an example, python's numpy has a regression function called polyfit. With a linear regression model and a moving window you will truncate the beginning and end of the dataset, but that shouldn't matter if you have a lot of data.