I am buying my first property and paying cash for it and I was asked if I wanted to order a location survey.  

The day land is more expensive in De Moines Iowa than oceanfront well know people are no longer lying when answering surveys.

Get the survey done and get a lawyer to advise.

At the link below, there is a survey from University Presidents.

I don't think I'd consider any of that to be perfectly fine, you can check your survey that you may have gotten with your purchase to see if it is on your property, if it is I'd see if the neighbor is willing to split the cost with you. 

The seller and his attorney recently paid for a survey to be done and also paid someone to spend a lot of time at the city to persuade them.

I'm an Illinois real estate attorney and have little knowledge of the real estate laws and customs of North Carolina.However, Cash is king and you should make sure you are covered/protected before you complete the transaction.First, yes complete an inspection if you want to and that can take as long as you can to have it completed.Next, although I may feel ok with giving the Seller's attorney my Earnest Money, I would not do the transaction unless a title company is involved as the escrow agent for the parties, takes/receives my payments, and can guarantee me clear title ownership.In Illinois, as a Seller's Attorney, I can be ready for a cash closing in about a week, subject to the time it will take to obtain a Plat of Survey, HOA documents, mortgage payoff letter and city inspections, and cleaning up any title exceptions, if any.  

Although we weren’t changing the exterior, we were asked to provide a survey of our property, elevation drawings, stamped FAR calculations, and sat in front of the city councils to obtain an approval (the special permit).

@Antonio DeFlorioI've also seen:Buyer to pay for survey and provide a sealed copy to seller.Buyer agrees that their attorney has reviewed the contract and accepts all terms.All contingencies waived.Should seller decide to appeal previous year's property taxes, buyer is to pay their pro-rata share of the legal expenses to do so.Like you said - buyer pays all closing costsAnd all of these things are always buried in the middle of 16 pages.

I am currently surveying the NC market for a 3bd 2 ba home that is predicted to appreciate over time.