If an HOA wanted to implement policies with regards to screening tenants, how long is their leash?

Investors (usually from DC and NYC) come to Baltimore frequently seeing these $17K “treasurers” and usually hear back home broken and defeated 6-12 months later.Last point, for anyone looking at Baltimore, lead paint is a (very) real issue. 

As did the petty thieves who felt like it was a piggy bank just waiting to be broken into. 

They cannot "screen" the offers.

The first repair he wants to do is to fix the 3 ceiling fans throughout the house. 2 of them don’t have power going to them and 1 is broken.

Broken. 

We did not return the deposit - when we added the damage, unpaid rent, broken window issue and late fee on rent - it exceeded the amount of the deposit.2.

Often employed by professional tenants to mislead mom and pop landlords.For me it is a obvious red flag when screening.

Today they want to move because the apartment was broken into.

If your screening standards have a set minimum this can not be negotiable otherwise you have no standards.