How to appropriately test Ethernet hardware?
I would check the rise times of the 25 or 50MHz signals and make sure they conform with the rise times required in the datasheet of the phy. An RMII shouldn't be that difficult to use since there are no tranmission lines involved. Make sue the PCB has a ground plane and the 50MHz signals are on the top layer (with the ground layer below) for best results. If you really wanted to go crazy with testing you could check all the RMII signals with a logic analyzer and make sure there is no errors, but this would come with a great cost in time.

Checking the TX/RX lines is more difficult. To measure the lines you'd need a differential probe (200MHz but ideally 500Mhz or more, diff probes are not cheap) to measure the differential signals of each pair. Some scopes provide test patterns (with additional cost) to help make sure the eye diagram conforms to the timing standards.



Source: http://ww1.microchip.com/downloads/en/AppNotes/AN2686-Ethernet-Compliance-Test-10BASET-100BASETX-1000BASET.pdf

Should any of the above be sufficient to say the Ethernet, in
electrical terms, is good in my device? What should be the criteria?

Noise can cause timing to be off. Too much EMI caused by other sources on your PCB can cause a signal to change from a high voltage state to a low voltage state, this would show up as bit errors in ethernet packets (or worst case a register on the phy to be improperly set if the error is in RMII). I think the criteria should be make sure the phy is properly configured, and packets can be sent from your device to another device with very little error (like losing less then one in a million or billion packets)