I'm curious as what y'all consider to be the minimum criteria for approving a tenant, assuming you don't have access to professional tools.

Or did you get approved for a certain amount before you went to auction?

Is there a specific duration I should be pre-approving an inquiry within?

Still, you may not have to submit your tax returns. more info would be needed to know for sure, reach out to a seasoned loan officer for pre-approval and see if your situation would allow for this. 

I also require additional security deposit and sometimes a guarantor if they were approved in this manner.I also agree that this situation sounds like a pattern not a one-time mistake or short term tough situation that caused this.  

To help expedite the rental unit approval/inspection, you can also talk with your local housing provider that runs the Sec 8 program what basic items they look for and make sure you have those covered. 

Your goal is to plant the idea in their mind and create a situation where selling to you feels like the easiest and most logical solution for them.If you haven’t already, the first step is to get pre-approved for a mortgage so you know what you could afford if you were to purchase the property conventionally.

Currently in the process of getting my business visa approved to make the move south (from Canada).

Credit and everything else is good and I'm already pre approved, have an agent to work with, and the market identified. 

Here are some ways to get started analyzing deals.For a Single Family House Hack:-Reach out to a couple lenders to give you a good idea of what you are approved for.