For roboshow we have a pretty great setup (and Full HD) but its not nearly as inexpensive as you are looking for. But here are some things that will translate to what your looking for.
First the audio, pretty much every mixing board has an
AUX send, you can pick which channels, and how they are mixed, and sent out of these ports. We used this at Robocon once YouTube took down our stream for copyright (you live you learn) and it worked well other times we did it. On smaller boards I have seen them labeled as Monitor out.
For encoding we use a Mac mini and two devices to get full HD.
We use a
BlackMagic Ultra Studio Mini
Its supposed to take embedded audio but there are some serious driver issues with that and losing sync, so we use a
Focusrite Scarlett 2i2
What great about this device is it can use XLR or 1/4" inputs, with individual gain controls and monitoring on the device, so if we have to make a long run for the audio cables its not a problem finding long XLRs and the lights around the knobs blink to the audio so we can see we are getting signal.
As i said before we have used YouTube, but also uStream and LiveStream. Nothing really to say about any of them that hasn't been said. Unfortunately for free there aren't many options.
With that said we had a streaming partner
Stream Monkey that gave us super awesome streaming service for the Orlando regional.