you could use target disk mode to migrate the files you want from 1 mac to the another. http://support.apple.com/en-us/HT5872 -- Move your data to a new Mac

Try reseating the LCD ribbon, because with everything else it sounds like the hardware is working fine. The issue is not getting the screen to wake up. Have you tried a hard reset while it put back to together.... its when you press the power & the home button at the same time for 20 seconds while it is plugged into a wall outlet for charging not a computer.

There might be a stuck key or a keyboard malfunction that make the macbook think those keys are being pressed down. Besides replacing the keyboard you could open the keyboard viewer to see if the keyboard registers any keys being pressing. If the are pressed down the keys turn gray. You could always go into the keyboard preferences and disable the keys that seem to be malfunctioning. Of course if this doesn't work you can alway use an external keyboard and disable the internal.

Try reseating the heatsink on the processor and a lot of times the immediate shutdown is due to an overheating issue. The processor only take about 3-5 seconds to get up to 300 degrees and if there is nothing cooling it down it will shut instantly to protect from further damage.

Try preforming a hard reset on the device. Plug iPad into power outlet from wall NOT computer, then hold down the top power button of the iPad and the front home button near the button. the Apple should display itself then you can let go of the buttons.

CAMERA Primary 5 MP, 2560 x 1920 pixels, autofocus, LED flash, Features Video-calling, geo-tagging, smile detection Video 720p@30fps, Secondary 1.3 MP

Make sure when you replace the screen that no dust or debri gets in the enclosure and that it is sealed properly...also the ribbons connecting the lcd to board may also be faulty. you can try reseating the screen again and then test or if you are still under the warranty for the part used you can see about getting it exchanged.

"This accessory is not supported by this iPhone" Usually means that 3rd party device you are using has not been certified to work with you product. This does not mean the the 3 party item will not work with your iPhone just that the interaction with the iPhone may has unforeseen complications. Most docking stations and 30 pin connector etc.... may not put out the right mount of juice to supply the iPhone. The Message "This accessory is not supported by this iPhone" is not always a bad thing to see, but if your dock connector starts randomly disconnecting a sync or stops charging intermittently i would discontinue use.