Quote: Originally Posted by
DuoMaxwell 
This LCD screen runs off of 12V DC so I shouldn't have any issue hooking it up straight to the battery.
That bit worries me. The screen might be 12V, but your battery is 12.6V and with the engine running, it's 14.4V. If you do that to the screen you've just bought on eBay, your $150 will not last long.
Also, if you're running the amp and the laptop off the same wiring, it'll mean the same fuse. Laptops aren't power hungry, but that and the amp would be enough to pop a 10A fuse if they both turned on at the same time. The PC really should have its own dedicated power line and fuse for safety reasons, and just share the ignition-on cable as that has no significant current draw on it.
Your non-sound problems could be some sort of remote-on signal between your non-existant head unit and the amp - a simple test is to touch the audio jack ends when you think it's on, to see if you get a buzz from the speakers. If you dont, you may have to replace the amp too or find a more complicated solution. That's proprietory gear that you'll have to search the forums for your answer.