about your tv :
the DVI connector is PERFECTLY AND TOTALLY FINE ! DVI _IS_ HDMI (without sound and hdcp crypting and some color space which not concern us, for your purpose : DVI = HDMI !)
DVI = HDMI
so, if your tv gives "weird" picture, you have to check :
- your mac use the correct resolution (1080p or 720p )
- your tv is configured to give full-pixel (the whole thing the computer is giving, no treatment, no weird video transformation or...