For the hdmi (remembering I know only a little about hdmi 
) I think this is what you want (noting the warning about the two output devices needing to be the same which may affect you):
It would probably be best to ask the folks that have done this what they used to split the hdmi output. Some of the screens you showed earlier have hdmi input, one of those may be as cheap (and less hassle to set up) than an hdmi splitter in to a vga converter. As to the Arduino there are 2 on board regulators, one which takes the 9 to 12 v from the coax barrel connector on the board (which goes through a diode for polarity protection) and then does appear (but without the diode for protection) on the pins as Vin. You could hook your 12V supply to the Vin pin, but it would be safer to solder wires to a barrel jack (similar to the one from the DC supply that the 12V plug that powers the video) on to the plug where 12v comes out. Then that barrel jack plugs in to the jack on the Arduino where it is expecting to see 9 to 12V coming in which it polarity protects with the diode and feeds to the 5V regulator (which in turn feeds the 3.3V regulator) to provide the 5 and 3.3 V power supplies in the Arduino (using the jack also shuts off the 5V power from the USB cable if it happens to be plugged in as well which is a good thing too).
Peter