If she insists on paying you cash, then she needs to hand it to you, so that you can in turn write her receipt for her records.Tenant #3:   Sounds like she's trying to negotiate the rent without actually saying so.    

The tenant insisted on moving out, so the owner agreed to let them out of their lease without having them sign anything.

You need to insist that dogs be on leash at all time and fully under control of the owner.

And if they ask you point blank for the name and contact info for the owner, you just say, "Sorry, she insists on staying anonymous."  

The landlord vouches for him and insists he is perfectly capable......

Lady does not want to move and insists that I tear down the kitchen and the AC closet.

And insist that they pay for the damage to the skylight.Are you at market rent?  

We insisted they come and walk the property with us as it was not to our standards.  

But, keep in mind that once a tenant starts thinking that the place needs a bunch of work, they will usually get picky.I learned this when applicants would insist on seeing a unit before it was ready.  

But this neighbor thinks the insurance process too slow and insists that my tenant or me to give her credit card number  so that she can use her favored contractor to fix the support beam in next 48 hours, otherwise will file lawsuit against landlord.