I would deny based on whether they had an outstanding debt/judgement to the prior landlord.

Do to the state mandates that require 100% reserves in all categories, combined with the rising cost of their neighborhood insurance policy, my strong recommendation is you find out what further anticipated spikes in association dues & if any possible special assessments are looming.

For example, I mention that every adult must pay a $30 application fee, pass my credit/criminal background, make a combined income of 3x the rent, and whether or not the property accepts pets.

There is a company called Fund & Grow and they help you navigate this process and get between $50,000-$250,000 in combined business credit lines.

Using mortgage debt at 6-7% on a refinance to pay off higher rate debt such as a credit card at 15-25% is a savings!

If you don't have the extra cash, can you get creative with a private lender or partner stacked on bank debt? 

The properties are rented and the debt is reducing via tenant rents.  

So is this just as simple as showing how much the property makes or do they also look at personal debt to income, credit score, reserves etc as well.

The auction will often take place at the property although these days that's usually combined with an online auction that starts up to 2 days before the final bid is accepted. 

Therefore, you don't need to use a 1031 exchange; you can take as long as you wish to find and then buy another property.The only exception is if there is debt on a property, in which case there will be some UBIT (Unrelated Business Income Tax) implications.