I had appointments with my handyman to do some work, she was unavailable.

I don't normally do appointments anymore.

But our repair guys are okay with scheduling appointments with the tenant then billing us, so you have to have that in place, too.  

I have scheduled with my current tenants a day to show the house to several interested people (Open house by appointment on one day rather that having to bug the tenant a bunch of times).

I haven't risked anything except canceling appointments with people who wanted to see it. 

A $10.00 processing fee will be charged for each payment received either in person or by mail or by any other method.NO SMOKING:Smoking is not permitted inside the home or garage.Applicant or an appointed representative named by letter must physically view the property prior to submitting the application for rental.

We have all been sick, and have had several doctor appointments over the past year relating to coughing/headaches/sinus infections/etc.

I'd contact SmartMove about that.But yeah, I always google people, before I schedule appointments usually. 

A "caretaker" is generally exempt from license requirements, assisting one person, justified by her age, health, abilities and without compensation with some personal relationship, but, laws of agency, meaning principle and appointed agent make you responsible to her as well as in how you represent her.

Time to update your lease to say "cash is accepted in person by pre-arranged appointment only".