If you have equity, income and good credit, any lender will do this.

I feel they've set low standards.

The child's father is trying to reduce his child support down to a ridiculously low amount.

Some bids were crazy high while and I decided to go with one that was very low. 

There is government data online to see where good schools are, low crime rates, and desirable city amenities.  

So, have the mortgage note signed by the person whose credit I checked who is an agent (owner?)

Local credit unions or banks could provide funding that is more suited to your requirements and have a greater grasp of the community.Understand the local construction codes, zoning laws, and any other rules that can affect your ability to make improvements or rent out the property.Make sure your escape plan is well-defined.

While your immediate position doesn't change, you now own a property with 200k of currently inaccessible equity.Also with your first one, if you can put down 5% or equally low, occupy it, and do a live-in flip, you may not have to refi; save up for another 5%, do it again, new primary loan, etc. 

Typically 30Y fixed, refi into an LLC, low doc loans that don't report on your credit. 

You have to calculate based on a more than a few things like loan amount, down payment, taxes, Insurance, (Any HOA) PMI, based on credit, etc.If you are 1099 the Bank can look at the most recent 1099/Schedule C if you have been in business for 5 years or more with a 25% ownership in company or more.