Denying Applicant - reasons

[Tracie Van]

I am going to deny a rental applicant based on them not checking the agreement box about falsifying information on the application, and giving permission to contact employers, landlords etc. 

Additionally, they did not provide proof of income as outlined in the applicant criteria provided. 

I want to know if I should even mention that his behavior while inquiring and applying was rude and showed dishonesty. The unit was just vacated and not listed, and this applicant contacted me. I told them I had a unit coming up, but we needed to do work in it and get it ready. He/she texted and called nearly everyday, numerous times asking all kinds of questions and when it would be ready. I told them they needed to fill out an application first and I mailed one. 
I was contacted again and he said he was going to be in the area and wanted to look at it. I told him that I had two DR appointments that day, and could not show it. 
That evening I get a text from him saying he saw the apartment and its great! When will it be ready. I see a VM from him; he said he went to the apartment and the painter was there so he went inside to look. I was furious. 
Next text, when is it ready? I said I just received your application and need to process it. The unit still needs to be cleaned, carpets cleaned etc. He said that's okay I have friend who can clean it and I'll just vacuum and it will be fine. Extremely pushy, and not even approved yet. But I could already see I had reasons to deny him as listed above.

Do I even mention that he violated our trust by letting himself into the unit when I told him I could not show it, or just use the incomplete application and no proof of income and be done with it?

Thanks in advance