Apparently they hadn't had an inspection in a while and had bare bones coverage.

The mortgage company will probably require that you carry a policy every year and will tell you how much coverage to purchase.

I suggest to my clients to carry water backup coverage and personal injury. 

It includes, among other things, a comprehensive map of the Cleveland area, grading each area from 'A+' down to 'F', using a consistent, objective and transparent methodology using publicly available data.

If so I would be suspicious since such policy coverage is based on driving record, credit rating, age, etc.

How much are you setting aside for expenses and debt coverage? 

Property owners are still required to have insurance coverage for the rental properties.

If the closing was say 10/1/19 and the policy was issued but the closing did not happen until 10/14/19 (not unusual) often the policy is not reissued with the correct date.Technically, there is no coverage because the Insured (buyer) does not have an insurable interest until the closing date.

All of my other properties I have insurance coverage for replacement cost, but on this property I am considering getting only cash value coverage.

It is also important to note that some insurance agents are bound to certain higher levels of coverage than others.