More than likely the LCD in the display was damaged during the drop. The force of the impact on the corner of the phone probably damaged the LCD. The colors changing and spreading on the screen and then eventually the whole screen going black is definitely a sign of a damaged LCD. You will need to have the lcd or the whole display replaced to fix this issue. As far as the battery goes it does not store any of your save files. It is not a storage device for files. Your saved data is stored on a flash memory chip which is located on the motherboard. So you do not need to worry about loosing any data when replacing the display or battery. It will still be saved in the memory chip on your motherboard. Now if your phone was damaged to the point it won't power on and the problem is a damaged motherboard then one should be concerned about loosing saved data.
So a new display should get your phone back to a fully functional device without any data loss.