What is the first Navy Enlisted/NCO Professional Development course, and what is generally taught there?

What happens is after you find out you are advanced to the next pay grade, your command will set up a time/day to do Petty Officer Indoc; the basis of how to learn to be a leader. After which, you are frocked (meaning you wear the rank but you will not be paid as of yet).
For E4-E6, we don't have a "school"; when you are an E7 or higher, you can elect to take a Leadership school (from my understanding). When you attain the rank of Petty Officer, your senior enlisted look differently at you. While you aren't doing grunt work majority of the time, they expect you to dish out the grunt work and deal with the complaining.

With counseling, are you talking about reprimanding or when we, as the "new" leader, needs assistance with a decision or to get an option from higher leadership? Since I've been a petty officer, my professional relationships with senior enlisted and officer has been very good; they have put a lot of trust into me and with what I can do with my job description. If I'm not able to assist, I do my best in researching a way for the task to be completed 150%.

Every promotion takes passing a course and exam and having appropriate marks on evals

When I served there was a basic Petty Officer Indoctrination, but nothing really to develop true leadership skills except for senior petty officers. I knew of a couple of E-5 that enrolled in the sergeants course on the Marine Corps base.