I have to do more research about hand held scopes and power quality analyzers. What little investigation I've done so far seems like these are a little out of my league in both price and skillset.
Yes, they're expensive and much more complex than a DMM, but they will give you useful information about the output of an inverter, like how close it is to a pure sine wave (power factor, distortion). I'm not sure how useful a DMM will be for this, at best it will show if the voltage is close to 120V and if the frequency is about 60Hz, but not if it's a square wave (sorry, modified sine) or sine.
Yes eventually I will need a clamp meter as well, but I would use the multimeter before connecting the mains to check which outlets are on the same circuit in the same or adjacent rooms when using house power for example.
How would you measure that? You can measure if they're on the same phase by measuring voltage between them, but in that case you'd be measuring voltage, not current. The only way I see of verifying if they're on the same circuit is loading them until the breaker trips, and testing if the other outlet is also without power.
The reason why I mention clamp meters is that some clamp meters have also some limited voltage/resistance measurement features, so it might save you from buying a DMM.
As for separating the neutral and hots how would you suggest doing that without opening the cables because I cant just do that on rented equipment? I suppose I could make my own mini-20A rated extension cord but it seems impractical to make and carry around a whole bunch of them.
Yes, you will need some sort of extension cord / adapter. Note that for a DMM, you need something similar, since you have to put the meter in series with one of the live/neutral wires, which is much harder to do safely.