I would highly recommend you do a full housing replacement. The glue that holds the glass on is extremely difficult to remove. You'll be chipping away at the glass for hours and probably ruin the frame.
You don't want to use iPhotos to import. Instead, you should use the new Photos app. Press and hold the alt/option key when opening the Photos app, and it should let you choose a location to import from. Select the old iPhotos library from the drive. Photos will then create an updated copy of the library that will also be shared with iPhoto to save space.
It is most likely not water damage to the entire logic board. Probably a small amount of water from rain, or another source got into the phone. Since the water damage indicator near the battery was red, the water probably got into battery connector. This would short the pins in the connector causing rapid power cycling ultimately killing the battery. Even though replacing the battery, in this case, would fix the problem, I would recommend inspecting the logic board for any other corrosion before continuing use. Doing this will ensure you don't have any issues further down the road.