Since it seems that we have had to follow up on everything (a good habit, we think), including the finalized HUD, what other important documents, etc. should we make sure that we have on hand?

Get or give a digital copy to/from the other party once the document is returned with it's stamp. 

Everyone should know how to acquire documents in their farm area and that can be done in a couple hours usually just by going to the courthouse and tracking things down within that system.Names of offices may be different too, as Jay mentioned the county survey office, we don't have one really that's in the loop, it begins at the Assessors Office getting the tax ID Number, from there, with that number you can get the plat map at the Planning and Zoning Office, with the plat number you can get the book and page of deeds at the Recorder's Office.

I paid a listing fee for legal documents and that was it to get started.

But later when my agent wanted to see the approval, she dismissed it and we found out it was only a receipt from the bank for the seller's documents. 

Consider documenting the entire process in a thread here or on a blog like J & Carol Cheers 

2) how do I protect myself (as far as what type of documentation should I demand to see, what should I be named on, etc.)?

I do hire cleaners even when the tenant leave its "broom clean" because new tenants expect better than that.If you have documented repair costs for things like the cabinets and countertops I think you can charge for those.  

If you have enough documentable income, you can afford to two carry houses (your primary and a second house or vacation home). 

When I checked the documentation that was sent by the firm, it was only addressed to the owner and not the lien holders.