I have a 7th gen civic and as I am sure you know, sun is our enemy. Even the cluster of guages in the dash gets washed out because ofthe angle on sunny days. Screens will be no better.
As for the screen question about the battery, this applies to all screens: Yes regulated. You can run it straight off the battery and it will work and actually be a bit brighter, but it is not good for the screen. The voltage will fluctuate from ~6 to ~14 volts and the 12v stable it needs. You can run it Battery --> regulator on cord --> screen or psu --> screen (bypassing the corded regulator).
And fuses are your friends too.