cord movement shutting off the electricity is a definite fire/shock/electrical short hazard that needs to be fixed immediately. A bare wire could be contacting the box or another wire which is why it shuts off.
1. the paper and plastic in the box would be a fire hazard.
2. the boxes are supposed to be deeper than the outlet to hold the wiring behind it and must be a minimum size according to the number of conductors in the box to avoid heat buildup.
Sounds like you may have plaster walls, the ears broke off the box, or the box not installed flush with the wall among other things. They make a plaster mud ring for attachment to boxes that extend out a 1/4" or so which are usually used with plaster walls. Otherwise the box needs to be moved out. For further safety, install a GFCI outlet.