It's an ingenious circuit, although tricky to get it operating right.
The resistor at pin 6 of the 4013 is a good idea, to make sure it will be pulled low initially.
should I also be able to decrease the "on" time by decreasing the size of the resistor? Is there a point at which increases and decreases no longer has a significant effect on the duration?
As post #8 mentions, the on-time is affected by the neighboring resistances around the capacitor. A 1000 ohm value for R12 might result in 1 second. A 10k might result in 10 seconds. Notice the capacitor is also discharging simultaneously through D3 and the solenoid.
Here are some direct tests to make.
Try putting a meter across the 100uF capacitor. You should be able to read the rising charge on it.
Also read the voltage at pin 6 of the 4013. You should see it rise as well.
The above tests are done while the relay is energized.
After the 2N3906 shuts off, the capacitor discharges. You can read how fast by using the meter.
The 1N4005 is there to absorb a high V spike coming from the solenoid at the instant it is shut off. At any time was the diode absent when the circuit was operated? If so then components could be damaged.