My beloved Nexus 5, died on me Friday so I picked up the nexus 6. So far I like the phone only 2 small issues. I can't get OK Google to do anything without unlocking the phone. On my nexus 5 I didn't have this problem. Also the other issue is sometimes when I'm making a phone call and remove the phone from my face the screen doesn't turn back on it just stays off till the other person hangs up. Thanks in advance.
Go to menu>settings>language & input>voice input>enhanced google service (tap on gear icon)>"OK Google" detection and make sure always on is selected.
As for the screen staying off, i recall some early issues with the proximity sensor when the N6 was first released, but i never saw any of the problems. Did you buy this phone new or was it second hand?