You might have to solder the screw to something else in order to be able to take them out. It goes without saying that you must buy a replacement screw set for your computer.

Have you tried to reinstall the firmware? Apple is offering the download for your model on their website. https://support.apple.com/en-us/HT201518

A bluetooth antenna is not very high tech. I don't believe there is any chip on the wire itself, so the computer won't "reject" the new cable. If it doesn't work, you can always buy a USB Bluetooth adaptor.

This is the procedure to reset the SMC from Apple: To reset the SMC: Shut down the computer. Plug in the MagSafe or USB-C power adapter to a power source and to your computer. On the built-in keyboard, press the (left side) Shift-Control-Option keys and the power button at the same time. Release all the keys and the power button at the same time. Press the power button to turn on the computer. Give it a try, let us know if it makes any difference.

I don't see why I won't work. However, keep in mind that the system might be very slow or unresponsive if you boot from a external drive through USB 2.0. You should consider replacing your optical disc drive for a hard drive, this way you could use the SATA port.

A usb connector is made of even smaller connectors. One of them is power and that seems to work according to your description. Have you tried with a different cable? There could be a short in the cable, specifically on the "data" wire.

Have you tried connecting an external monitor? Usually when there is problem with the hardware that the system can detect you will hear beeping, in a sequence according to the issue found.