1) When DUT is under mains power, and you are using a handheld scope, since there should be no path to ground, this is the safest method for probing and signal tracing with having to worry about what you touch even when the scope is chassis grounded to the DUT?
2) If this is true? Does the situation change if that scope is powered by a DC wall wart? I assume no....am I in error?
1: As long as you are working on the low-voltage side, and being very careful. Do it as your own risk!
Always use a 10:1 probe, and with either the clip end or grounding ring protector on it, so there is no chance of the ground band on the probe touching anything by mistake.
Only connect the probe ground clip while the equipment is off and discharged.
And use the classic rule for live equipment - One hand only, near it. Keep the other behind your back!
2: Not worth the risk. You are undoing the ground isolation. At best, most wall warts couple noise to the output & at worst it may cause a fault.
Part of the vagueness in answers is that your question could apply to high voltage circuits as well as low voltage.
If you are strictly working on things with less than 50V power (after the AC mains transformer or PSU), and a chassis that can be grounded via the test equipment without problems, then you can use any type of scope.
Just keep the probe tip grounding ring covered to avoid shorting anything via the scope.
If measuring non ground referenced voltages, use two probes and set the scope input in differential (subtract) mode. That will display the difference between the two signal points, without a floating ground.
And do not wear any rings whilst working in anything you need to put fingers in to! They can cause accidental shorts, and provide a lower resistance path for high voltages making any shock worse.
Someone I know almost lost a finger due to his wedding ring shorting a high current low voltage supply it and instantly heating enough to cause serious burns.