It SOUNDS like a battery issue. I've seen that with iPhones as well. When the battery goes bad or is faulty it jumps around in %'s and then can die randomly. Not sure about the logic board, but I can't imagine that's not bad.
It sounds like your LCD may be damaged (hence no display but everything seems to work). Pretty common issue with liquid damage. Sadly it is a seriously expensive part.
we have seen this issue before with the new (2012) model ipad 2 power flex cable. What we did was just simply use adhesive tape to hold the cable in a position that all buttons/functions work and then just tape it down in that position so that it stays there. It has worked fine. Those "jawbone" connectors are finicky sometimes.
Yes it can be. It is the same as all other models. Some colors, white for instance, is adhered better than others and is more difficult. Not sure why. But the basic construction of all S3's is the same. You can heat the screen and remove just the glass.
The connections are hard to get on properly when closing up the iPhone 5. I had a similar issue on a repair before but reconnected the cables and was very careful closing up. Try that.
The reason why the corners don't sit right is because the rear housings that are available are aftermarket. I actually stopped providing rear housing replacement in my repair shop because of this. I could never get them to sit properly. It does sound like the LCD connector is not seated properly. Have you tried a hard reset after closing the phone up?
I have had something similar happen before. The phone would power up and work fine until I threw the back on and then once the metal prongs on the bottom of the back touched the midframe to screw in, it would shut off. Try changing the battery. I cannot remember exactly what corrected my issue, but it bugged me to no end. It was a customer's phone. LOL Eventually did figure it out. Sorry I can't remember for certain what it was...
Is the LCD glued to the front glass/digitizer?