Applicant threatened me to return application fee

[Ping W.]

One applicant thought I went overboard on screening them and asked too much for pets has threaten me trying to get their application fee and application form back. I have run credit report on the husband but not the wife. I have written comments on their application form while screening so I did not respond to them. The wife text me "If we do not hear from you by end of business day tomorrow we will be contacting the appropriate authorities". Have I done something wrong? How should I handle this situation? What they could do to hurt us? 

I only rent out our own second home. I am inexperienced in this business. Please help!

Thank you in advance for any advice you may give!