Can we proof a scavenged transformer of unknown specification?
Since the wires are 16 AWG can we assume it's safe to run at up to 10A? Or could it be internally more constricted? 


Based on apparent dimensions, the current rating is almost certainly less than 10A. Measure the laminations carefully and estimate the rating by comparing with published data for similar transformers.


  Can we "proof" it for a current X by running it at X for a few minutes and confirming that voltage and current don't drop significantly, and that its temperature stays below, say, 40C?


Such a test would take more than a few minutes. Winding temperature rise should be determined by resistance. Search "temperature rise by resistance." You will need to select an appropriate and conservative value for maximum temperature rise.


  Is it safe to run the transformer in reverse (step-up): I.e., can we connect the output to 120V AC and take 720V off the other side?


Absolutely not. You can use it in reverse by applying no more than 20 volts to the 20-volt terminals. More than that will cause the transformer to saturate, draw too much no-load current and overheat.