I have  a home I rent and the lease states no pets, but allows one dog per a pet addendum.  

There's a ton of info on my website if you need more info.Also, a pet peeve, is when investors are "sold" into very expensive "asset protection packages"with fill in the blank forms for all of the above entities at these big real estate sale pitch events.

I am not to discuss rent, pets, deposit or other lease related terms with people who are looking to re-rent my unit.

4 pets?  

My big concern is that this applicant has a service dog and I never allow pets.

Pet deposit was received for pet.

You'll have to decide what criteria is more important to you to use as a "tie-breaker" criteria - higher income, higher credit score, no pets, etc. 

Vacancy, turnover, and not enforcing your lease (late fees, evicting, documentation, pets, inspections, violations, and more).

They don't think about pets, their affect on common areas shared with other tenants, and the landlord's responsibility to provide safe common areas. 

You may want to consider testing the carpet for pet urine.