I noticed there is no pet policy currently in place.

Talk to your insurance agent.Tell him you have your first rental and want a home owners policy tailored for this purpose.Every landlord I know who has been in the business for any length of time has a 1 million dollar umbrella policy.

I have two policies.

I buy lots of insurance and have an umbrella policy. 

Bigger companies may have different policies. 

I believe it is to show proof they aren't paying more than HUD says they are allowed to, otherwise they are indeed in violation of HUD policy.

I would suggest looking around the east side of Salt Lake.This "Judgement map of Salt Lake" is eerily accurate:http://utahpolicy.com/index.php/features/today-at-utah-policy/2814-check-out-this-hilarious-map-of-salt-lake-city

I had my insurance agent quote me with a 6-month Builder's Risk policy, and we're cancelling the seller's insurance policy (which is currently with the lender!

We also would like to avoid them being named as an owner to prevent them from any liability (though an umbrella policy may be the solution to that).

Both through their professional activities and through your ability to go after them if they fail to do so (they carry E&O policies for a reason.)The buyers' agent doesn't cost you anything, he is paid by the seller.