Would love to hear suggestions for providers that work well with STR managers, and if you think General Liability Insurance is enough to protect against lawsuits in the event a guest gets injured at a property I manage, or damage occurs to their personal property as a result of a leak, fire, etc..

If someone injures themselves and sues, they will be suing the LLC and not you personally.

If it is mandatory and if we don't install it and if our tenant got injured by fire, then I think it can be big problem. 

Also realize the liability you create for yourself if a tenant or their guests get injured by your half way handywork, regardless if it happens to be plumbing, electrical, HVAC, or carpentry.

@Helena Wu a Premises-only Umbrella policy protects you from lawsuits from anyone on your property that is injured and blames you.

If someone injures themselves and sues, they will be suing the LLC and not you personally.

Then employer refused to provide seller (who is an injured party) with any further details.The seller can make this an issue if they choose to.  

Well, I wasn't going to have my tenants injured by the tree (was overhanging my deck area), so I decided to bite the bullet and pay for the removal myself (at least, the neighbor didn't make a fuss about granting access, etc.

The tenant injures himself/herself and tries to sue us and can make a case that we wanted to wrongfully evict them and we didn't have a right to?

A tenant who is injured by your negligence will sue you personally - your LLC will not protect you.