What's the proper procedure here in Florida?

It trains the tenants to follow our procedures NOT theirs.So, you could have the few actual emergency calls handled by a trusted family member or friend and make it worth their while to do so:)

The property now cash flows nicely and I plan on repeating the same procedure with another home. 

Just make sure you still have robust screening procedures and policies in place.

Now that I moved to another State, and can't go to my properties on a moment's notice, I see how easy it is to eventually lose my gains on capex if my PM doesn't stick to procedures, can't/won't go themselves to see important issues as they arise, and after they are fixed.

It has nothing to do with whether they care or not, it has to do with their investor guidelines or internal policies and procedures for short sales.

They will NOT audit or otherwise verify the data I submitted to them, they do NOT perform procedures designed to discover irregularities, should they exist.

(CRMC).These are companies that have been audited by the National Association of Residential Property Managers and have demonstrated a high standard for both procedures and customer service.Again, I believe the best way to find a good property manager is to seek a property manager out and has invested in advanced training and education and has attained designations, usually they also have years of experience.

The question is purely procedural: is an escheat considered to be an inheritance tax payment or  does one need to explicitly pay it off?

It's not unreasonable since numbers in the South are going up, but did your agent explain the procedures and they can do once a week showings within a two hour window and everyone will have masks and gloves and only one group in at a time?