I know the pictures make it difficult, but where is the refrigerator?  

If it's your refrigerator, then it's your responsibility to ensure it works properly. 

She never put anything in the lease that I was to keep her stove, refrigerator, or furniture as I threw them out to use mine.I've looked everywhere on line and I'm worried she's gonna Sue me for her abandoned things because I feel the judge is on her side since he was rude to me.  

I supply fully applianced homes, washer/dryers, dishwashers, refrigerators, range.

It's a little incentive for them to clean the stove, refrigerator, etc or they will have that deducted from their deposit.Best of luck to you

I do have a list of criteria (mostly financial and criminal history related) I've posted on our refrigerator so that people will know prior to applying what we are looking for. 

I am trying to rent my first property should I get a new or used stove and refrigerator or leave no appliances, what about a dish washer, garage door opener, or garbage disposal.

You have an undercounter refrigerator.  

She left basically everything in the house except personal belongings... food in the refrigerator, dishes in the kitchen, etc....

The first estimator pointed out that it's obvious the reason the old unit isn't working, the refrigerant lines have been pulled off the unit (and sadly all the old (Freon) refrigerant has escaped, polluted the atmosphere).