Hello,
You can test for max current by applying a known current and testing for temperature rise. With some diodes though they have to be soldered into a PC board of the right type and has the right pad sizes and possibly heat sink copper.
You can also test the forward voltage vs current this way too.
To test for maximum reverse voltage, you apply a reverse voltage and look for a maximum leakage current. It's nondestructive as long as you do not go too high on the voltage, which means you have to monitor the current.
To test for light sensitivity, you shine a bright light on the body and check the photo current. Clear glass diodes will exhibit more sensitivity than others as they act like a tiny solar cell.
Sometimes you might want to check the capacitance too.
To check the recovery time, you have to push a current through the diode in the forward direction, then suddenly reverse the applied voltage. You can check the time of the short spike of reverse current as the diode recovers. Some diodes have zero recovery time, but it may be just very low.