I don't leave a W/D on my flips and I'm also tempted to stop leaving a fridge too.

If the fridge has water-through-the-door, it may have a small tank in the fridge to cool the water; you need to empty that tank, or move the fridge somewhere warm.The furnace, if new enough, might have a water drain that operates even when the *heat* is running. 

The people who were renting out the side I'm going to be renovating/occupying left a bunch of stuff in it; couches, mattress, food in the fridge, trash in various places, etc.

I was also going to put a mini-fridge down there.

The fridge was new and they were the first to use it.

Is not including a fridge common on your area?

During the great downturn a few years ago, brokers/realtors were forever removing fridges and stoves from bank owned properties.

Since all of the major capital items were replaced shortly before or after purchase, I don't foresee anything on the capex side over the next 5-7 years (maybe a new fridge but I'm ok rolling the dice on that one). 

Is that small black thing a fridge?

My duplex or if im looking at one, I try to have separate utilities to have the tenants pay for them so they can have as many fridges as they like!