Would it be possible to convert DVI to HDMI (version unknown), then use a HDMI 1.4 cable to a TV which was created for HDMI 1.4? Consider my "scheme" below: (m = male, f = female)
DVI (m) → HDMI (f, v.?) → HDMI (m) v1.4 → TV (f, supports HDMI 1.3)
Yes, all you should need is a (passive) DVI-to-HDMI adapter and your cable to connect the PC to the TV. I have even used a cable with DMI connector on one end and an HDMI connector on the other. The digital video does not need to be converted, just use the proper connectors.
For the audio connection, you may not have to do anything extra either. At least on nVidia GPUs, two channel (stereo) audio is also output on the DVI connector (when enabled through the nVidia control panel). Otherwise use an analog (line out to red&white RCA plugs) or digital audio connection, although most (all?) TVs do not accept digital audio input. For multichannel audio (e.g. 5.1 or more) you would need to connect the audio from the PC to an AVR or pre-processor rather than the TV (which usually only accept stereo).