They used mule hats when they had overhead trolly lines for electric mine motors. In 1911 they didn't have electric in the mines???????? The mule hats don't make sense.
The doors were for ventilation. You had intake air and you didn't want it to go out the main door.
The kids opened it many times as the full cars where going out of the mine and when they took new ones in.
The plank off of the door, made it easier for the kid to open and close it. When they closed the door with a lot of volume of air, it would slam shut. The plank shouldn't have been removed, but I guess someone had pity on the young kids opening and closing the doors.

Is this the end of a shift? They wouldn't need 4 guys to lead the mules, so I am guessing the guys at the rear are about to knock off for the day. What do you think?

The mine was ventilated by a forced-air system in the vertical shafts. For it to work properly, the horizontal shafts had to be blocked off. More info here.

I wonder what purpose that door serves? It's obviously not airtight, it doesn't really look like it would keep anyone from going in or out. And they really need a kid to sit there all day just to open it when someone leading a mule cart comes by? I don't get it.