Electronic – use any HV Probe for a Voltmeter/Ampmeter gauge

I wouldn't expect that adding an LED would affect the meter's accuracy.

However, I would suggest trying the LED backlight idea first, in a temporary fashion, to make sure that you can really add an effective backlight. If the display was not designed to have a backlight, it may not be possible to add one.

1. Probably none. The 'scope is measuring voltage between the probe and the ground clip, which you had across the exact quantity that you wanted to measure.
2. Probably because either your 'scope or your circuit is floating/ungrounded. Both need to be grounded for that to be a problem. Good power supplies from AC mains will always isolate the circuit being powered and will thus require re-grounding as desired.
3. Probably because the sensitivity is identical between instruments, as it should be.
4. Probably because the meter, the 'scope, or both have a DC offset that is incorrect. 'Scopes are known for capturing fast signals, not so much for DC accuracy, though they're usually pretty good. If you have a low impedance meter, that could also throw off the reading, but it would probably affect the dynamics too if that were the case.

The reason that you're "supposed to" use two probes and a math function is to handle the case where both the 'scope and circuit are grounded and you want to measure something that is not referenced to ground.

For me, I keep my 'scope ungrounded and be careful of what I'm getting into. All of my controls are plastic (but there are often unused BNC connectors), and I only ever use one ground clip unless I have a really good reason. That ground clip then becomes my reference for everything. However, that only works for relatively slow stuff because the loop area between probes can become an RF antenna.