I'll provide some google terms for your request. Note that you have multiple orthogonal criteria, so any balancing act across those areas will "win" at different points for each person's personal opinions.

WRT ultra low power computation and ignoring everything else, google TI MSP430 family or Microchips XLP (extreme low power) PIC series. Typical marketing figures are double/triple digit microamps of current per MHz of clock at a couple volts, so figure double/triple digit microwatts per MHz (and usually tops out at not many MHz), which is of course fairly meaningless when compared across families because its a clock freq. Its rather important if a simple binary add takes one cycle at 300 uA/MHz vs 324 clock cycles at 30 uA/MHz, the better marketing figure does not necessarily provide best system performance figure of "actual real world work per microwatt"

If you want easy to use, that ranges from some $12 dev board from TI which does approximately nothing you listed, up to the somewhat famous Dimitry Grinberg emulator where he hand wired some dynamic ram and numerous other devices as I/O devices to an ATMega 8-bit chip, then wrote an ARM emulator on the ATMega, and stuck linux on the emulated system. Thousands of instructions per second. But hand wiring is often not seen as convenient as "open dev kit box, plug into USB, all done".

http://dmitry.gr/index.php?r=05.Projects&proj=07.%20Linux%20...

Finally WRT "general purpose linux" I'd personally define that as something with enough ram and storage to run some modern stuff on top of a JVM. Most of these specialized "linux" SBCs are around the hardware level of a early/mid 90s desktop and have the hardware to make a great router, or wifi access point type of appliance, but you're not going to run a modded minecraft server on 8 megs of ram, even if thats perfectly adequate for a home router or a wifi access point or a DHCP server or DNS server or whatever. So you're getting openWRT to act as a router, not a "mainline Debian install" Although there are small machines like the raspi that are almost but not quite "mainline Debian installs" and home-theater PCs like zotac zboxes that are just small desktops and therefore make great little mainline Debian boxes (for like 10 watts and $300, admittedly)

There are also interface issues such that for J random hacker a wanna be settop box is probably a lot easier to set up and operate than a wanna be network appliance. Solely easy or difficult at the I/O initial install level, not beyond that.

There's also definition problems with "low power". At a dollar per watt-year, there's economic noise level of low power where an old desktop is probably fine, even if run 24x7, and if you're doing this for an employer they'll be paying you about ten thousand times as much as a cost of employment as the cost of the electricity you'll be using in this experiment, or done at home it'll be "about a ten thousandth your income" so energy cost is not a major economic driver. Low power in that context merely means it won't have an effect on the local HVAC situation. Then there's deep space spacecraft low power, where every watt of electricity costs multiple kilograms of plutonium so all that matters is its reliable and low power and nothing else matters. Everything else falls somewhere in between.