I’ve filed suit and the court is scheduling a mediation via Zoom first.  

The fact is if you’re ok, hand on heart, to stand in front of a mediator or judge and justify your deductions against the bond - even if they disagree - you won’t come across as disingenuous and you will be confident as will your attorney in putting forward the case (should it go that far)

How can I be the best mediator and get these people to behave like adults?

It might go straight to mediation.

Consider the fact that signing a lease with each of the 4 tenants separately and independently will be very time consuming, and will likely put you in a position where you will be expected to mediate disputes between tenants. 

First did you use a board of realtor contract almost all of them have mandatory mediation for disputes like this.a very good RE broker will know your position if you asked for a credit they denied it I simply think you say OK I am going to close then. no harm asking.

It further requires that the lease be attached to the complaint and requires mediation if one of the parties asks for it.

Yes, LL has tried to mediate the situation without success and now, tenants must call the police to further access the situation and no doubt animal control will issue fines. 

The tenant upstairs and this tenant downstairs do not get along and we've had mediation meetings with no luck in finding peace between them.

The corresponding property damage involves the complete loss of the existing structure w/o the possibility of rebuilding.We are looking to get into mediation/litigation asap...and are hoping to find a law group that works on contingency in the Raleigh Durham area.Thanks in advance for any help you can give