Here (and I'm assuming in most places), the condo association governs STR restrictions, not the local/county governments.

-Are there legal restrictions on this in NC, such as the length of the option?

Some areas might have restrictions on renting portions of a home, especially basements.

Last week, the odds were 86%/14%.Dec 18 FOMC Meeting: 74% probability that rates will be 50 bps below current levels (a 25 bps cut in each of the Nov 7 and Dec 18 meetings). 25% probability that rates will be 25 bps below current levels (one 25 bps cut and one ‘do nothing’ meeting).They Said It“A market requires both sellers and buyers.

Quote from @John Underwood: First thing to do is check with the city or county to make sure the ADU would be permitted to be a STR.The municipality has an application process but initial indication is that it's not overly restrictive in the area I'm looking at. 

No restrictions or caps on supply, and I don't expect the City to revisit this anytime soon as it was about a 3 year discussion that I think council is happy to put in the rear view mirror. 

I don't beloved they have a unit restriction for owners

What groups are out there that lend to practicing Muslims and honor their financing restrictions?

And are there any restrictions on what you can use the “equity” LOC?

Unfortunately, such tenants are not commonplace.To maximize the odds of always having your property occupied by a reliable tenant, buy properties that attracts a tenant segment with a high percentage of reliable people.