The length of time needed for recycling/recharging the flash depends on the power source, the circuitry and any safe-gaurding circuitry which may prevent over-heating.
If a flash has not totally discharged-a full power/full duration pop-then it can be recycled fairly quickly. Mastery of exposure settings and some decent existing light are all that is required. Sorry, I'm not going to give you a set of master aperture/shutter/iso settings---they don't exist. Such usage is extremely hard on batteries, flash tubes and circuitry--hence the need for protective circuitry.
Of Pentax flashes, only the 500 and the 540 can be powered externally (by design). This extra capacity is often deemed necessary for rapid recycling/recharging. The 540 has a timer protection circuit---it will NOT rapid fire as you desire. The 500--not suitable for newer K series 100,110,10 cameras can rapid fire on a PZ1p, 6-10 times in very rapid succession-depending on exposure settings and needs.
In short your answers are: (1)NO (no(no)). (2)Won't do it, nothing to change.
But: The 360 using a lower power output: 1/4 or less, will fire very quickly for several pops. Achieving a lower power setting can be done manually or via the automation with appropriate aperture/shutter/iso settings and decent existing light. It's strictly a YMMV situation.
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