I'm sorta winging it here. I know how to design fast switching circuits, but I am no expert on reverse recovery time measurements. It turns out, as you might expect, that the techniques are different for small-signal diodes, e.g., 1N4148, than they are for multi-amp rectifiers. I don't think I want to tackle the high-current testing. Are you good with a relatively low current measurement? I think you will still get reverse current (you demonstrated it in the waveforms you posted), but the results probably won't agree with the datasheets.
Are you doing this out of curiosity, or for some explicit purpose?

 

I'm doing this just out of curiosity. I want to ultimately get a feel for device parameters.

I've got a good fluke MM, it has a resolution of about 0.1mA.

Here's some more transistors I have:
2n3904 (npn)
2n2906 (pnp)
2n5129 (npn)
2n4125 (pnp)
NTE312 (n jfet)

 

I wonder if the germanium diode lights like an incandescent light bulb if its forward current is too high?

 

I wonder if the germanium diode lights like an incandescent light bulb if its forward current is too high?

Probably for a few milliseconds.
I wonder if the inside is evacuated, or filled with nitrogen, or...?