They "paid you back" not because they knew they were foul playing, but to mitigate their business loss due to you not honoring your part of the written contract. 

What kind of blanket type of insurance I should buy to mitigate risks of any type of potential lawsuits from this problematic tenant.

This might reduce the pool of interested renters, especially for families with adult children or multiple adults sharing a residence.In summary, requiring all occupants aged 18+ to fill out an application allows you to vet the entire household and mitigate risks, but it could also complicate the application process and cause pushback.

Insurance probably sent an inspector who gave a ballpark repair cost in addition to the cost estimate the clean up company provided.I'm assuming you used the clean up mitigation company the insurance suggested, who may not be your best choice.Get another estimate to return the property to full glory, cost for paint, glue, dumpsters, every nitty gritty drywall screw, plus putting your tenant in like kind housing.

Other stuff you mentioned should be able to be mitigated but also should have been brought up what sounds like a lot sooner. 

If so, I assume I am obligated to mitigate through my contract, but I declined to allow them to do the work anyway.

On termination of the tenancy, property or money held by the landlord as prepaid rent and security may be applied to the payment of all rent, and subject to a landlord's duty to mitigate, all charges as specified in the signed lease agreement, or as provided in this chapter, including the amount of damages which the landlord has suffered by reason of the tenant's noncompliance with section 33-1341.

Building in a contingency to your budget can help mitigate getting in over your head.2.

Only way to mitigate the unknown is to assume a rebuild cost.

However, this might expose your operational LLC to risks associated with your rental properties, which could be mitigated by keeping them separate.Cons: Separating Y from X could provide an extra layer of protection.