Can I use newer, faster ram from new iMac in 2009 Mac Mini?

It will not work, the bus on the old Mac is too slow for it

No, Mac are very picky about RAM and it is best to closely follow the spec's when replacing or adding modules.

It may or may not. I upgraded the memory in my 2013 Mac Pro (1866 MHz DDR3 ECC) and the removed memory works fine in my 2009 Mac Pro that uses 1066 MHz DDR3 ECC.

It only runs at 1066 as expected.

Your analysis was almost spot-on wrt to the Voltage. We are working with Dual Channel. The speed here is not an issue at all, as it moves the data much slower than the design speed of the modules. There remains thus the Latency and the Voltage. The timing {with CL7 at 1067MT's (533MHz) and CL11 at 1867MT's (933MHz)}, could also not matter that much, but I don't have access to a Lab at the moment to measure that exactly for you. Meanwhile you can look at the table at the bottom of https://en.wikipedia.org/wiki/CAS_latency to see generic timings involved.. I strongly suspect that, with the lower frequency being used than designed for, the modules might just work and not overheat with the added 150mV supply voltage. Good quality RAM will take that. Note that it has been tested to provide at the rated voltage, Random Access at 933MHz(half the dual rate of 1887MT's) and then only with a Latency of 11. Lower Frequency can provide a lower latency, but higher frequency could add to the latency. It is however not an engineering forum, but a users forum. See http://www.hardwaresecrets.com/understanding-ram-timings/ for more on the subject.

Remains only the form factor: If the pinouts and key are the same, it might just be worth a try. I know that I would, but then I have a free hand to do just that.

Leo