Op-Amp Comparator Testing
If you want to test the comparator itself, and not the fact that it is configured to have additional hysteresis, there are a few tests you can do. If you want to verify the hysteresis you can apply a slow triangle wave to the input and measure at what voltages the comparator trips high/low at. This is called a  (VTC).

Voltage Transfer Characteristic (VTC) plot for hysteresic comparator

Probe comparator output and input, put scope into XY mode
Apply a low frequency triangle wave.
Observe and use cursors to measure the hysteresis.


Load Regulation

Apply an input to force comparator output to be a constant high/low
Measure voltage at output of comparator with no load
Attach varying loads and measure voltage at outputs (be sure to not exceed abs max current!)


Rise and fall time characterization with different loads

Probe comparator output, and trigger on rising or falling edge.
Apply a large signal square wave, DC centered at the trip voltage.
Zoom in to the rising edge and measure rise time (10-90%, maybe 20-80%)
Zoom in to the falling edge and measure fall time (10-90%, maybe 20-80%)
Attach different loads (common is 2kΩ || 1000pF) and repeat


Propagation delay

Probe comparator output and input, and trigger on input rising/falling edge.
Apply a large signal square wave, DC centered at the trip voltage.
Measure the delay from 50% mark on input to the 50% mark on output.
Do this for switching low to high and high to low.


Max toggle rate

Probe comparator output and input, and trigger on input rising/falling edge.
Apply a large signal square wave, DC centered at the trip voltage.
Increase the input square wave frequency until the comparator output stops toggling

Make sure you have good bypassing for this one.

As for getting more data, you can repeat these tests at different supply voltages and common mode voltages (common mode being the trip voltage).
For the prop delay, if you have a good function generator/scope, you can measure dispersion of the comparator. I would remove hysteresis for this test. Here you change the input voltage so the input just barely trips the threshold. If your trip voltage is set at 2.5V, you apply a square wave of something like 2V - 2.51V (500mV underdrive, 10mV overdrive). Measure prop delay, and increase the overdrive to 20mV. Repeat. When the prop delay is plotted against the overdrive voltage, there should be a y = 1/x looking curve.