"What is this part"に関する回答The replacement screen you are installing probably already has that piece. Can you take pictures of the new screen you are trying to install?
"Can’t download, update or delete any app"に関する回答I don’t have a lot of experience with jail-breaking but by the sounds of it something is definitely wrong with the firmware. I would probably start by wiping the device and trying to flash a new OS onto it?
"My Phone won't turn on!"に関する回答So based on the info you have provided I would say there are two possibilities. 1) The battery is dead. This is entirely plausible, assuming you have not replaced the battery before, the battery would be several years old. Lithium ion batteries degrade over time, and the more they degrade the more likely they are to just fail altogether. 2) The phone experienced an unexpected shut down as a result of a weakening battery, which could compromise the operating system causing it to not be able to boot. So there are number of steps to take in order to troubleshoot the issue. If you are repair savvy, have the tools and the parts to do it, the first thing I would try is a new battery. It is more likely that the battery is simply dead than the OS being compromised. If you don’t have the tools or want to try everything else before going to a shop, you can try booting into DFU mode. To do this, you will need a computer with iTunes, and ideally an original apple cable. Here is a guide for booting into DFU mode...