RFM69 and Ethernet shield share SPI bus, each one with its own CS line. Ethernet shield uses pin 10 so JP1 should use pin 6 for the radio. Issues sharing SPI bus seems to be due to bad SEN line handling as described here. The official Arduino ethernet shield suffered from this, but most commonly used clones work ok. Shields including 74LVC14 as in the pic should work ok.
I've confirmed Julien's results. I procured a board that's inferior to his but with the same chip on it, and with no load other than its LED it can remain operational as low as 15mv. With that as the input voltage, the output voltage is around 640mv, so I think the reason my board can't survive on less than 15mv is that when the output voltage dips below about 600mv then the chip no longer has enough voltage to switch. Compared to other off-the-shelf chips, this is an impressive result, and even more so considering that these chips only cost around 8 cents each in small quantities.