Second -  is there not recourse for the original criminal fraudulent action this woman took by forging signatures, etc?  

When you say "background check", do you mean the legal/criminal history report, the credit report, or both?

If there is a police report about the assault and a conviction or a guilty plea, Alabama law says it is a non-curable default if there has been a criminal assault. 

You won't be able to run a credit/criminal because there's likely no history on them.

If the investment has professional administration, you will have access to the answers to those questions, but also much deeper vetting of the sponsor that will include personal background checks covering things like bankruptcy, judgements, litigation, criminal history, and other red flags related to a sponsor that you would want to avoid, plus other items that investors rarely consider like ensuring that the properties are actually titled correctly, property taxes are paid, and insurance is current.

Many criminals avoid places if they know there is a chance of getting caught.

It's hard to describe but after dealing with so many criminals you kinda get a feel for people.

Since you didn't mention it, I'd also recommend you do a criminal background check with a signed (real or digitally) application that gives you permission to do so.

Criminal histories-- Prior evictions-- Criminal convictionsCredit / Payment history-- Is your credit score at least 740+-- Have you filed for bankruptcy recently?

Also, your estranged wife did not legally "trespass" as she is co-owner, and even without giving notice it's not a criminal offense for her to be there.I get it, have been divorced (more then once) myself plus decades of call like this on ABC (you pick subject) and it all boils down to the same thing.