The part that your looking for is called a signal wire cable or a service antenna. Here is the link for it: Nexus 5 Antenna Cables Hope this helps!

It's probably an LCD damage. Go on Ifixit.com and follow their troubleshooting guide corresponding to your phone.

Go on Youtube and look up the different ways to re-boot your android. Or follow the "I forgot my password" process and go from there.

IFixIt it's an online shopping store, troubleshooting guide, and step by step phone or other devices repair. I recommend you to buy a screen that best fits your phone from Ifixit.com and follow step-by-step the replacement process. It's not that difficult I promise.

I recommend to restart your phone if you already haven't done so. If that doesn't help, call your voice mail and select "delete all messages" option. If the problem persists call your phone company as it might be a network error.