Can they even do this and should I get my legal on this or just ignore and rent anyways? 

I am exploring my options for borrowing against a property that I am the legal title holder to.

That may not even be legal in most States, just get a licensed roofer.... 

Also, you must familiarize yourself with local laws and regulations regarding STRs to avoid purchasing in areas that are not STR-friendly, which could save you from legal headaches and poor investment choices.Another critical strategy is staging your home to appeal to your target guests.

The PM company has it on their website that "If you have been evicted from a property, we consider this a legal event, and the application will be denied." 

I am posting looking for some guidance and possible advice (as much as legally possible) in order to help a family member.

Can I ask the seller to legalize the unpermitted work before closing?

In the world of development you have money at risk to complete these and other reports, legal, zoning, permitting, environmentals, etc before you go hard on your deposit with a seller.

From my understanding, the legal obligation is a 7 day notice for a month-to-month lease in North Carolina.

However, if you want to go after the balance and you have an eviction judgement from the court, you can higher an attorney to find legal means of collections, such as garnishment.