If it doesn't exist as an add-on that you can easily install, I know that XBMC has a web browser (or has an add-on for a web browser I can't remember) so you can go to the website and watch the episodes there. I don't know if it supports flash though and using an ARM-based cpu is out of the question if you need flash in a browser.
Not true! Although there was never any Flash player released for ARM Linux by Adobe (and projects like Gnash just don't work for the encrypted media content streams), they technically did so in a roundabout manner through Android development, which is actually why I recommend Android on ARM for a media center platform. It can be hit-or-miss with the specific ARM CPU, and Adobe isn't officially providing support or security updates for the platform anymore (much like Linux x86/64), but they used to maintain Flash for Android up until this last summer and certified the packages to run on Android up to 4.0.x. You can still download the APK installers for manual installation from Adobe's archives.
http://helpx.adobe.com/flash-player/kb/archived-flash-player-versions.htmlhttp://www.adobe.com/devnet-apps/flashruntimes/certified-devices.htmlDoes this mean you open up your shiny Android device to a potential security risk on the general internet? Yeah, it does. But if you're using the device solely as a self-walled garden media center and refraining from using the system to access any sketchy websites for content (or avoiding most web-surfing in general) and sticking with official media channels, the risk is pretty low. This is why XBMC being released as an APK for Android is kind of a big deal now. All the parts are in place now with minimal fuss to make a really solid homebrew media center with a cheap ARM device that handles everything from Netflix to PBS to MLB from official media content provider apps from the Android market, and everything else in-between with XBMC's web browser and Flash support. The need for Windows as a backend to get all this content is no longer an issue.
Q1: What's the best solution for generic (non-Netflix/Hulu) Flash streaming (like from thewb.com, cartoonnetwork.com, PBS.org, etc.)? Is there anything that will aggregate TV episodes from these sorts of sources and present them with a nice UI?
Khao already answered this one, and it's the same solution that keeps being brought up:
XBMC.
Q2: Anybody have hardware recommendations for a lowest-cost (including low power consumption) MythTV backend capable of transcoding to MP4 (or Theora) from a dual-tuner HDHomeRun? When I tried to spec one out, it seemed like it required $300+ for an Intel x86-based solution, but I wonder if there's an AMD APU, or better yet, ARM setup that could do better.
Rummage around for a headless Windows laptop (broken LCD screen) that matches your power needs on both wattage and CPU requirements. A fully functioning, refurbished and venerable Lenovo Thinkpad T61 should be plenty on its own, and you can get
those for under $200 these days.
Here's one on Ebay for $80 that only needs a couple parts and a bit of love. They're not hard to find, and they're incredibly Linux friendly. Between having the screen off and manually throttling CPU performance down, you'll probably get the lower wattage performance you're wanting out of nearly any laptop CPU.