Those boxes are usually indicative of a bad connection to the LCD. Where the bad connection is would be anyone's guess. The cable could have been strained where it attaches to the LCD, or it could have been bent, folded, stretched along it's length. It could also be the connector on the ribbon side, the connector on the logic board side or simply both of them where they meet. You may want to replace the LCD to get new cabling. If that doesn't fix it you are likely going to be dealing with logic board damage.
This issue is because of a firmware issue. The point of manufacture did not flash the appropriate firmware to the LCD. You need to replace the LCD with a new one - but maybe from a different vendor so that the chances of the problem repeating itself is reduced.
Have you installed a new digitizer? If you have you will probably need a new - higher grade version. Also check the back of the cable for solder pads and exposed metal. You may have to cover some metal but that won't help with the lock/volume problems.
Don't count on replacing just the glass. The part is cheap yes, but you will need many more tools, including a method to cure new adhesive. It is a very involved repair and if you are like me it might take a couple broken LCDs out of your first batch to get the hang of it. I have it worth my while to buy the LCD assembly with the small parts preinstalled. Makes the installation much faster and almost completely painless. A total of like 9 screws and four plugs (I unplug the battery).