Or the buyer could just miss a notice that the loan has been transferred to a new loan servicing company (as they do, loan servicing companies are terrible IME), and miss some payments by accident, again negatively affecting your credit.

Everything I've read about it sounds horrific.

We would be so grateful for any input, experience, or recommendations from homeowners or agents regarding what would be the preferred choice for 1) attracting rental guests in the short term, 2) retaining value for resale and not looking dated in 10 years, 3) protecting ourselves from accidents.  

Many sleepless nights over how we’re going to pay our NYC rent (read: horrific) once she stops working... hence putting all of our savings into this little 4br 1ba with a $300/month cash flow (conservative) and hoping to build up from there.

And fined retroactivally.Additionally, if a tenant has an accident (slips/ falls/etc), or something goes hugely wrong with the dwelling (fire/electrical/etc), your insurance WILL NOT COVER the claim.

For the most part, if your running real estate somewhate properly, the risk isnt your property causing you to get sued....Think of it as you are hopefully protecting you property from yourself in the event a motorist sues you over a car accident, you dont want them getting access to your real estate assets. 

I know some people that have property free and clear don't care if they lose it and only carry liabliity insurance to protect against accidents on the property, medical or legal bills, etc.Do I have no other recourse but to carry full coverage?

If it was an accident-storm, crack in foundation, etc, I would do the regular checks and rent to them provided you are fine with the pets.

You may also want to confirm that your estate planning is in order to ensure that someone can seamlessly step into your shoes to deal with tenants in the event of accident or death.

Accidents happen and grills and fire increase those odds but that is what home insurance is for that peril.