Hi,
A/D converter, especially the high speed ones are energy eaters. Do not worry, it is normally to get hot. As long as you can see the image on the screen, you may assume that you have a working adc. The contrast and brightness do not affect the power consumption of an adc. Its power consumption is mainly dictated by its sampling frequency, not by characteristics of the input signal. Voltages seem to be ok. Regarding the passive components: in extremely few cases these componens fail and generate such defects like in your case. My guess is that you have to search for defective active components. The input switch you told me about can be the cause, but not necesarily. If the image have the same aspect regardless of the input you use, the chances are low. Anyway, you can try to bypass it in the following way:
-use the oscilloscope to identify the inputs and outputs. put a totaly red image on screen, then see with the osc where it gets in into the chip and where it gets out. do the same with green, blue and syncH, syncV (note that maybe synch and syncv do not enter into this chip, they may go into a digital processing chip). extract the switch chip gently, do not destroy the pcb under, maybe you will have to put it back. then make the bypass between red in and red out..and same with others. before starting with this, make sure that ic is only an analog switch. if the chip performs other functions over analod signal, this procedure will not work.
also, as a general rule, make a visual inspection of the entire board and look at the following components:
-power disipating devices (transistors from power supplies, coils, etc.)
-mechanical components (signal inputs, power inputs, adjustments)
sometimes these components can produce bad contacts between them and the main pcb, due to aging, vibrations, etc... if you see a poor soldering joint, remake it.
good luck