When I try to use my Bluetooth to initiate a phone call, the voice dial gets disconnected while I have the phone within my pocket or in the belt holster. If I take the phone out of my pocket or holster and hold the phone sideways in my hand, I can initiate the phone call. I have also noticed when trying to enter my password for the voice mail box the screen will black-out but I am still connected. I have turned off the knock-on feature without any difference. Is this something that is fixable without making a trip to the store to get a replacement: still under warranty. I am running lollipop on the device.
What BT headset are you using? Using one with Ver 4.0 is best. I use an LG HBS-750 with no issues. I am running 5.0.2 as well on my G2. The screen black out: I have my timeout display settings at 30 sec. You should also be able to either wave your hand over the screen or put the phone near your head to activate the proximity sensor and keep screen on. I also use Smart Screen to keep screen on.
The screen timeout is set for 30 seconds and smart screen is enabled. It blacks out while I am typing on the screen, which it should not do. I am using an LG HBS-730.
Sounds like you might have a digitizer/screen/sensor issue. If under warranty, have the phone replaced. You COULD try a factory reset to see if that helps with software conflicting with hardware.