Well, that's a new one. Have you verified that the headphone output exists in the mixer, and if set as the default playback device, will not output audio?
Re: monitor: I don't know the exact fix for you, but some monitors have odd HDMI input behaviors, due to HDMI largely being a TV input. Make sure that the resolution is correctly set to the native resolution of your monitor. If not, you need to try overriding that, somehow. If it is correct, then look for overscan/underscan options, somewhere in the OSD menu, and your drivers. I haven't had to deal with it in some time, but, Googling, "hdmi underscan radeon" got plenty of hits, even when trimming the time down.
Edit: it appears some monitors would, with incorrect EDID info, always overscan, so AMD defaulted to always underscanning, if the monitor doesn't underscan (why under/over-scanning didn't
die off, with digital connections, is beyond me). *sigh*
http://blog.galets.net/2013/02/fix-hdmi-overscan-without-catalyst.html