My Nexus 6 turns itself off randomly; How can I fix this?
This is a partial answer based on the output of logs provided.

Okay, after looking at the logcat, my guess would be a buffer overrun error originating/stemming from mm-camera and connected services/processes and ion memory allocation.

Snippets, 

08-11 17:51:19.504 E/libaprpmem( 1539):  Ion allocation success virtaddr : ret=2996989952 fd=48 heapid=33554432
08-11 17:51:19.507 E/libaprpmem( 1539):

08-11 17:51:19.332 E/mm-camera( 1539): mct_stream_destroy_buffers: Exit

08-11 17:51:19.337 E/mm-camera-isp2( 1539): isp_port_check_caps_unreserve:807 ide 3000f port 0xb6749d60
08-11 17:51:19.337 E/mm-camera-isp2( 1539): isp_handler_destroy_stream_param:1616 num_streams 0


Okay so don't be ready to quote me on that but dig this article discussing buffer underrun/overrun http://www.articlesfactory.com/articles/computers/buffer-underrun-and-overrun-scenarios.html

I can't pin it to a specific specific cause but try setting your camera resolution to as low a setting that you can get it, and if you can, disable your proximity sensor. Either when the camera crashes it takes your phone with it or even though it crashes , seemingly repeatedly, it keeps gobbling up your ram till your phone tanks and shuts off. Hit some development sites and see what devs have to say or are doing with respect to this subject if it's common. It could be "fixed" by a decent/corrected kernel build. Unless you are under contract or tied down by the knox/enterprise-ability thing.

You may need to go pirate/commando and break your contract to save your data though. Meaning rooting and ripping all your data off the device.

OH! and i forgot this,

08-11 17:49:33.932 W/ResourcesManager(29010): Asset path '/system/framework/com.google.android.media.effects.jar' does not exist or contains no resources.


That one is a genuine WTF.run ls -al /system/framework/ and look for zero length *.jar files.

More- missing lib

08-11 17:51:10.373 E/mm-camera( 1539): Tuning lib open failed: dlopen failed: library "libmmcamera_tuning.so" not found


Edit- Rerun your logcat "live" as logcat |grep F/ as before with phone connected till crash.