Possible options I have considered include:leave things as they are since things are going smoothincrease the rent and get sued if my research isn't correct increase the rent and keep the tenant month-to-month, with the risk that tenant may decide to leave and I may get an undesirable new tenant  along with costs to clean/paint and update the unit between tenants increase the rent and have the tenant sign a lease, otherwise move out.  

It may very well be, but I'd guess that this behavior is recognized as undesirable for lienholders because no one will bid the properties at the auction because this sort of behavior discourages it.  

I'm looking to begin investing in buy-and-hold properties in peripheral NYC markets: Brooklyn, Queens, New Jersey, Upper Manhattan. 

Google Craigslist ghost postings.all caps some keywords multiple postings with the same text will get ghosted meaning they let you think you're posting but it actually doesn't become visible.this is Craigslist attempt to weed out undesirable content to increase the quality of their content.

But if someone pinned me down with a gun at this moment, I would say A property is high neighborhood, high desirability, B property is good quality working class neighborhoods, C property is lower middle class and fringe neighborhoods that are either just declining or are just coming back, D property is lower class property in undesirable areas but houses are still standing, and F is war zone material.

Great pickup for you, very poor investment for him.We have this discussion at work all the time, and I have first hand experience where Australian's got sold turnkey properties in highly undesirable locations in Cleveland and they lost all their money. 

No point rehabbing and trying to sell something that was undesirable for other reasons.I'm not local to Jacksonville, but I studied it a bit, and that is not one of the areas I'd be looking to invest in.  

There were also some undesirable tenants on the property.  

I was initially hoping to invest here on long Island but after some thorough research have come to the conclusion that to have a positive cash flow I would have to purchase properties for section 8 in undesirable areas.

This helps me to quickly disqualify those sellers with homes in undesirable areas (at least for my company).