The first problem was with the video - no matter how I played with the graphics settings, a strip of the display at the top and bottom was missing from the TV picture. After some Googling, I found out this was a problem with overscan - apparently TVs can deliberately zoom into the picture provided by a video source. There is some more information here in MythTV's Overscan page. For me, the solution was simply to disable overscan using an option in the TV's menu (a Panasonic Viera HDTV).
The second problem was a distinct lack of sound through the TV. I've lost the source of this solution, but here's what I had to do:
- From System -> Preferences -> Sound -> Hardware, select "Internal Audio (1 output)", and then from the Profile section, select "Digital Stereo (HDMI) Output"
- From a console, run "alsamixer". Using the right arrow key, select "IE98 1" and then hit "m" to unmute this channel.