Where do I apply and what is the best way to screen section 8 applicants here in New York City? 

This will get me to at least consider your application.

In that case, the notices with attached fines would be the most you can do, but (and I don't know the applicable law off-hand here), you might not be able to start doling out fines if you didn't specify your notices would include them in the lease.

For example, for reasons we still don't understand, covid-19 created a run on properties, we have been flooded with tenant applications.

This is my first rental, and very first application. 

Does rentprep and zillow charge their fees per applicant or is that fee for the entire household?

I have had one potential applicant say they couldn't and let them know there are plenty of teenagers who would most likely be more than happy to do it for them but they are responsible for the cost which they were fine with but were rejected based on the application.  

I for one plan on holding my course and rejecting applicants with evictions on their record especially if it were actually enforced by the courts vs. just filing. 

You simply process the applications.

The application is all online and they do the background and credit check for $40 - charged to the tenant.