You may be confusing documents and patterns.An affidavit death of joint tenant would be recorded along with a death certificate but would not necessitate a probate to effect an insurable transfer.There are summary probates in some state that require minimum court involvement.In my state, CA, a formal probate is required when the Decedent's interest and their total gross estate exceed $150,000.

The tenant lease has force beyond your closing date - - in every state.Once you are the owner, you can proceed with various means of "removal".BTW: you should have made that stipulation in your offer;buyer is to occupy upon closingbuyer is to receive property vacant, all costs to be born by the seller.

Before you proceed any further, I definitely recommend getting 2 or 3 licensed general contractors to come out and look at the issues with you, and get some estimates on costs so you'll know what you're getting into.

I don't even know if I should proceed with anything else (background check, employment/rental history verification) if I feel weird about the types of jobs they have.

Evict them and get a new one as quick as the courts will allow it.

What if it isn't enforceable and the guy knows that and takes you to court?

I finally did a quit claim because my lawyer advised me it could be another year in the courts and more costs at which point, I'd be just proving a point rather than getting any of my investment back.  

With pictures and supporting statements from witness and police, your tenants could have gone to court to get a restraining order immediately.  

They haven't left, What documentation do I need to bring to court in order to evict these people? 

Daily Court Review houses the list for $19.99 (Harris County)...Along with Storage Auctions and Legal Notices, https://www.dailycourtreview.com/Step 1, you must register...Step 2, well connect with me and I'll give you the run down...If you go on the first Tuesday of every month, bring a lunch.