Yesterday, tenant found listing on furnished finder and made specific requests and they were hoping to get in that day, so I told them they could deposit cash in my bank and I would email the lease directly, which I did.

I average about $500/yr on average, and in most areas common wear and tear are not able to be withheld from deposit.  

4 tenants (students) were supposed to pay security deposit today for April 1 move in.

Keep the deposit?

Good tenants have options, they don't have to stay with a landlord like you describe who doesn't try to have a good business relationshipBy your logic, this tenant should have broken their lease when the refrigerator was not fixed/replaced within 2 weeks and gotten their deposit back.

We had one case where the sale was court contested and cost us $45,000 in legal fees and after 9 years of Depositions, Interrogatories, etc, we still had not gone before the judge. 

I would suggest adding a pet addendum or modifying the page of the lease that refers to pets.Having the changes initialed and dated by all parties and reflect the changes to accounting "pet deposit" or additional deposit.

First question is to ask them if they're approved today can they pay the deposit today?

My understanding is that so long as they confirm that they're moved out and give me the keys, I'm well within my rights to just have a junk remover / trash company take whatever was not there to begin with, and deduct that from the security deposit?

When I self-managed C Class properties, I would put the requirements for tenancy in the rental listing (3x rent in income, 1-month security deposit, criminal check, etc.), and then when the potential tenants called in, I would speak to them for a few minutes, explain the requirements, get some sort of verbal confirmation on some or all of the requirements, and then invite them to different showing times I had set up.