No you can't however you can get some aftermarket housings that are shaped exactly like an iPhone 6 but a smaller version for the iPhone 5. They're calling it the iPhone 6 mini conversion. I've done a few myself.
Sounds like water damage to me. You could try cleaning the areas affected by rust and corrosion with 99% Isopropyl alcohol. I would also highly recommend never to try and update or restore this iPhone or you will end up with error 53!
BSOD is usually related to long screw damage where by using the wrong screws on the LCD/Digitizer shield can damage traces on the board underneath the screw hole. You could also take a look at the tiny filters near the battery connector in particular FL11. Check that it's not missing or damaged.
You could have damaged the LCD or a connection may have came loose when you dropped it. Here are some basic troubleshooting solutions- 1. Hold the Sleep/Wake (power) button and home button together for a few seconds to see if the device powers on. 2. Try plugging the device into your PC to see if iTunes recognises it. If iTunes does recognise the device then this can suggest there may be a problem with the display. You can also try to update and restore the device via iTunes before looking into any hardware problems.
Try this: Make sure you are connected to wifi. Go to time and date settings. Turn 'Set time and date automatically' OFF then back ON. Then restart the phone by holding both the Sleep/Wake button and the home button at the same time. This issue often happens when disconnecting the battery from the phone during a repair. Update (01/11/2016): Hi Amy, your next option could be to try and restore your iPhone. This will completely erase everything on it so remember to back-up up your data before proceeding with the restore. Restore iPhone