Would my fridge wear or be damaged when I unplug it for a week?
Warning: A refrigerator unplugged with the door closed will become a factory of disgusting mold. 

The cure is to leave the door open.  However, it is impossible to keep other people from closing the door the moment they see it open.  

We have 4 fridges at a lodge, which we unplug to save electricity.   Every single time I unplug a fridge and block the door open, I return to find the door closed and the inside of the fridge a disgusting nightmare.  When I have asked these people "why mess with it", they said either habit, or that it was sticking out, or that it merely offended their sense of order.  

Electromechanically, the fridge won't care.  It will save ocmpressor wear, and also energy (though not as much as you think on a newer Energy Star model..)

In fact, the off-grid solar folks have stopped recommending super custom 12V fridges with 6” thick foam walls, and just say to choose well among the readily available Energy Star models,  and run it off an inverter.