the thing is that the TV set should appear in CCC as soon as it's plugged in, no matter the channel it's tuned to. in my case the TV shows up in CCC even if it's switched off (standby), I don't even need to turn it on.

Here is what I'd do:
1) completely uninstall the video driver, the HDMI audio driver and the Catalyst Control Center, in short all the ATI drivers and software pertaining to the video card
2) reboot in Safe Mode and run Driver Sweeper to get rid of leftovers
3) reboot in normal mode and reinstall the video driver, the HDMI audio driver and the CCC
If it still doesn't work, I'd try with a different cable and/or a different TV set (logically, this should be done before all the other steps, but since it probably is a problem to find another cable... for free... or take the PC to a friend's to try with a different TV set... if you can borrow an HDMI cable you should try that first).

Press F8 at startup, then choose safe mode from the menu.
If you come to a point that your TV set appears in CCC, then set the "extend desktop horizontally" option, select display #2 (TV) and start with a conservative resolution from the drop down menu, one which your TV can surely handle; set the refresh rate at the maximum allowed (probably 60 Hz); then work your way up to the max resolution allowed by the TV set (720p or 1080i or 1080p, worst to best, depending on the TV)