I do. I was just traveling about a month ago and the gate info had changed on the monitor but not on my phone and this was both ways. I always check the monitor and use that as the system of record over my phone.
I do. "The phone" is just another hassle one doesn't want to take care of on long-distance travel.
Paper tickets just work, can't say that for the login for travel / rewards apps when you think you were already logged in when being offline.