To be honest I have only tried local credit unions so far so maybe I need to go one level down from even that.   

If your W2 is your only means of financial security, you are standing on a very rickety bridge. 

I have about 130,000 equity in my house, but honestly my credit isn’t very good so I’m not sure a HELOC is a great idea.

Who's got the most competitive HELOC rates in Central Ohio for someone with excellent credit. 

I assume BP stands for BiggerPockets.

If NYC has the names/addresses of people - then they stand a greater chance of being penalized.From my understanding - NYC has already been penalizing those that the city has been deeming big players in the airbnb business and mostly let go of the small fish.

She is now a single mother of 4 with no work history, terrible credit and an $1800 mortgage.

If not, you can still go back to the seller and request those items be repaired, or more likely, a credit at closing for the repairs.If the seller won't offer a credit or repair it, certainly consider backing out.  

Do you have the tenants go through your rental application and require them to purchase the background check/credit check?

The consensus is to fix the credit.