Running the Reverse +12V -AND- the PC power supplies +12V together is not going to work - you can't splice them together because it means when the reverse lights are on, the PC is going to be getting unfiltered power, and when the PC is on, so are the reverse lights. It is also potentially very dangerous, depending on what else is hooked up to that line. In my circuit I solve this with two normally-open relays that have the Inputs both spliced to always-on +12V, and the outputs are spliced. This prevents current from going the wrong way up the triggering wires. You could possibly use diodes to accomplish the same thing, but you have to keep in mind they end up dropping the voltage, which may screw up your ability to use a cheap linear regulator on the LCD (and on the reverse camera).
Using a Normally-Closed relay to skip the reverse-trigger when the PC is on is already in the circuit (The middle NC relay)
My circuit also has a delayed-on relay for the exact reason you specify.
I think the only other thing you mentioned was powering the screen via IGN/ACC - That was my original plan, and looks to be the typical setup, but I found out that if the PC is off, I get a bright blue "NO INPUT" screen on the LCD, so if the PC was off/disconnected I'd get that for the whole drive (except while in reverse) - so I've rigged it up to provide power to the LCD when the PC is on or if the reverse lights are on - both connected to two relays to act as OR logic.