1--iffy on that one, depends on the LEO. To be safe keep it on the passenger seat or dash and immediately notify the LEO that you are OC'ing and where the firearm is.
2--The safest bet is to not OC in places that are off limits to CC. You really can't go wrong that way. There is nowhere you cannot OC, but can CC. You can OC in a bank, but most post anyway.
2b--Cary, NC restricts OC on sidewalks and Chapel Hill does not let you OC a handgun under 6". Check here for specific city info.
Or contact town hall for their ordinances.
City, county, and state owned buildings and their parking lots are generally off limits to carry. You can leave them in your car. As are city and county parks. I would assume public transportation as well, but I don't know for sure. Make some phone calls on that one.
3--Technically yes you could OC, but if you are asked to leave you must leave or be charged with trespass. I would recommend not spending your money places that don't let you carry if you can do that.
4-- No they only have to post buildings generally. Check Municode above.
Here is the NC General Statutes site. The vast majority of gun laws are in the 14-criminal law section, particularly around GS 269 and GS 415 areas.
And this is not legal advice, IANAL, bla bla. Do yourself a favor and double check anything you read when you are putting yourself in a potentially legal issue. Trust but verify! And welcome aboard. Read through the posts here. A lot of this has been discussed at length on other threads.