Graphics card can do with progressive or interlaced scan.
If your TV doesn't have the resolution, the option wouldn't even be available so don't worry about the specs.
Now how long is your cable and what kind of cable? I expect HDMI of course. Now cheap cables may or may not be able to carry a 1080P signal. Short cables like 6ft would have no problem but if your cable is like 20ft in length then you really should look into a new one.
Your cable could also be loose.