since the lcd, earpiece and volume control buttons all attach to the logic board in the same area, is it possible you accidentally disconnected the volume control flex cable. its the connector directly to the right of lcd
most likely whereas the rest of the phone seems undamaged the display was damaged by water. It could also be the logic board itself. I would first attempt a screen replacement to see if that would fix it.
its a setting to preserve battery life. It assumes if youre on the phone you do not need to use the screen. if you put it on speaker phone the screen should stay on. There should be settings to change it so the display doesnt time out