Let me answer your questions in reverse order.
3: State appointed is most definitely still a lawyer. Court appointed counsel work very hard, and a private attorney who accepts court appointed counsel has an ethical obligation to represent the appointed case with all the skill and diligence he or she would use in a retained case. Most of the public defenders I know are excellent lawyers, although they do often have very large case loads, and may not have as much time for your individual case as a privately retained attorney.
2. Intoxicated and disruptive is a different charge from public intoxication, and different still from resist, delay, and obstructing an officer. Resist, delay, and obstruct cna be any behavior that unlawfully delays or obstructs the officer from the lawful exercise of his duties, such as giving a fake name, running away from a LAWFUL stop, etc. Public intoxication alone is not a crime. But being intoxicated and disruptive is. Disruptive behavior is defined in NCGS 14-444 as: (1) Blocking or otherwise interfering with traffic on a highway or public vehicular area, or
(2) Blocking or lying across or otherwise preventing or interfering with access to or passage across a sidewalk or entrance to a building, or
(3) Grabbing, shoving, pushing or fighting others or challenging others to fight, or
(4) Cursing or shouting at or otherwise rudely insulting others, or
(5) Begging for money or other property.
subsection (4) seems to cover what you describe.
1. As far as Miranda warnings, they would only come into play once you are in police custody. And if you were in custody, and no Miranda warnings were given, this alone means little, unless you then confessed to unlawful behavior or otherwise provided evidence that was later available for use against you. If the officer observed unlawful behavior, and he observed all he needed to see for a conviction before he placed you under arrest, then it matters not that you were not given warnings.
** Please consider this- NCGS 14-445 provides that it is a defense to a charge of intoxicated and disruptive in public if you are an alcoholic. Aside from the fact that this could be a defense, it never hurts if you feel you may have a problem with alcohol to look into it. If you do think you may have a problem you might want to contact alcoholics anonymous, or other substance abuse services that may be available.