A longer run with a shorted input is definitely an interesting point. Kind of getting independent noise data that also include longer times (e.g. Alan variation curve). There is little need to do this with many parameters. The main interest would be longer integration times, though very long times are very likely internal averaging. So something like 10 PLC would be enough. For the voltage ranges, the 0.1 V range would sense mainly the input amplifier, the 10(20) V range would mainly sense the ADC part and the 100 V range and 1000 V range should not be very different, both having possibly quite some effect of the divider (unless they actually use different dividers.
For the noise I hope that Fluke did not include the same odd 0.1 Hz noise bump like many Keithley meters.