The safe mode generally happens when the buttons for power or volume up or down are stuck or damaged. So the phone goes into safe mode cause it gets a signal that the buttons are being pressed. You can try firmly tapping the side of the device were the buttons are located to see if it's a mechanical failure that you can loosen the button back to the unpressed position but often times it requires replacement of the switch under the buttons to fix.
The proximity sensor that's part of the front-facing camera and microphone flex is not positioned correctly in it's slot or the slot itself is not clear enough for it to detect your face. You can test by placing your finger over the top of the phone screen across the microphone and front camera, it should turn off and back on as you over this area, you might find that you have to hold your hand directly on the glass to make it happen, this would tell you the sensor works but its not seated correct so its not sensitive enough to see your cheek while on a call.
If you use COMMAND + V you will boot in verbose mode. I suspect you will see errors reading the hard drive repeated, this would indicate SATA or Hard Drive. Since both are used to store the downloaded Internet Recovery it would be effected. You should get a thumbdrive as Dan suggests to isolate that set of devices from the boot process. You could also hold down D and see if diagnostics come up. If so run the tests.
On the yellow ribbon that runs through the frame to the board there are two things to look for. 1. The red end of the ribbon is supposed to line up with the white line, no line up not the right part. 2. That same cable says the part number on it. The one I have in front of me that's not working for this phone says MDH-7562-r54w