Heres the deal,
Component = YCBCR color space
VGA = RGB color space
Now both Component/VGA inputs/outputs can send both signals, however they rarely actually do both, as its more expensive to implement.
Sadly, in the case of a cable box it is highly unlikely it will be outputting RGB over component, unless you find a magic service panel to make it switch by holding setup for 10 seconds while it boots, spinning in circles 3 times and chanting an ancient celtic charm.
This leaves your monitor, which is the more likely culprit to be able to take YCbCr over VGA. Ive seen more than a view monitors that do this, but your going to need to check your manual and cross your fingers.
Also Component -> DVI is likely never going to happen on any TV. At best DVI inputs will take analog RGB if you are lucky, but most only take digital.
However if you goto comcast or whoever your cable operator is and bug them for a new box you should be able to get a box with HDMI or DVI on the back, both of which can go into the DVI port of your monitor.