Here's a description of my understanding of how the VCO works. Here's a graph from the same simulation setup described above except that I have fixed the input voltage on the non-inverting input at 2V.

The VCO in the electric mistress has three basic states. In the graph,
1. one state is starting at 8.953us where the current through the diode (green) is zero and the inverting voltage (blue) is below the non-inverting voltage (held constant here at 2V),
2. a second state starts at 8.911us where the inverting voltage has exceeded the 2V noninverting voltage and the output voltage (red) begins to fall, and
3. a third state starts just past 8.925us where the output voltage (red) falls below the inverting voltage (blue) and the current through the diode (green) is strictly positive (the green bump).
Here's what I understand to be going on in theory in these three states:
1. When the inverting input is below the non-inverting (blue below 2V), then the comparator output pin is allowing the output voltage to grow towards the positive rail, pulled up by the 10K resistor R4. The pattern is exponential. The actual difference in voltage at the inputs does not matter, just that the inverting voltage is below the non-inverting voltage.
a) There is no current through the 1N4148 because it is reverse biased. As a result, two sides of the VCO circuit are operating independently.
b) The current flowing through the pull-up resistor at the output is going into the output pin of the comparator. It appears that the comparator output is behaving like the lead of a capacitor connected to ground and the voltage is growing exponentially.
c) On the other side of the circuit, a constant current charges up the 47pF "timing" capacitor C1. In the process, the current source is raising the voltage of the inverting input. The charging of the capacitor linear because the current is constant.
2. In the second state, which is quite brief, the inverting input voltage has risen above the noninverting (blue above 2V) and the comparator switches to providing a ground for the circuit so that the voltage built up at the output during the first stage falls rapidly.
3. The third state starts when the output voltage drops below the inverting input voltage (red below blue) so that the diode becomes forward biased. The capacitor C1 discharges through the 1N4148 diode and that causes the inverting voltage to fall below the non-inverting again (blue below 2V), reverse biasing the diode so that its current returns to zero.
Now the process returns to state 1.
The reason this is a VCO is that the higher the input voltage at the non-inverting input the longer it takes for the inverting input side of the circuit to rise above the input voltage so the period of a cycle gets longer. Because the non-inverting voltage goes up linearly, the period is related linearly to the voltage input. And because frequency is the inverse of the period, frequency falls inversely with the voltage input.
I think the importance of the diode is how much it allows the capacitor to discharge and that this is primarily affected by the capacitance of the diode. The more discharge, the longer the capacitor charging time.
The VCO amplitude can be affected by frequency. At relatively high frequencies, there is less time for pulling up the output and so a lower amplitude. At lower frequencies the oscillation is rail to rail.