Can bitcoins be exchanged for nonlicensed "Professional" services? Can Bitcoins be exchanged for professionally licensed services by persons who are NOT licensed? For those folks, receiving cash would normally be a Blackmarket Racket since only licensees, the governments approved vendors, may partake in those business activities. 

Persons can dispense all sorts of professional services as long as they aren’t for "business" or "pay".  Good hearted family, friends, neighbors and colleagues do it all the time. It’s a basic freedom we as Americans understand, to not be controlled like individuals were in the USSR etc. 

Hypothetically, if neighborA were to dispense Legal Advice for PAY, yet NOT be a licensed attorney, then neighborA would be involved in an illegal operation according to political regulations.  NeighborA, however, could in fact give that same advice, and on any other matter they like...taxes, medical, legal etc., as long as they DON’T do it for business.

So, can these folks legally receive bitcoins?  Or, is that considered Pay and Business?
Persons can dispense all sorts of professional services as long as they aren't for &quot;business&quot; or &quot;pay&quot;

In some places, yes. In other places, no. Since you didn't specify your jurisdiction, I will guess that you live in Alabama. Alabama Code - Section 34-3-6 (b) defines the practice of law this way:

Whoever,
(1) In a representative capacity appears as an advocate or draws papers, pleadings or documents, or performs any act in connection with proceedings pending or prospective before a court or a body, board, committee, commission or officer constituted by law or having authority to take evidence in or settle or determine controversies in the exercise of the judicial power of the state or any subdivision thereof; or
(2) For a consideration, reward or pecuniary benefit, present or anticipated, direct or indirect, advises or counsels another as to secular law, [...] or
[2 lines snipped.]

For example, if you help your neighbor write up their will, and you weren't a licensed lawyer, you could potentially be in violation of this section, even if you never received any money.

Hypothetically, if neighborA were to dispense Legal Advice for PAY, yet NOT be a licensed attorney, then they'd be involved in an illegal operation according to political regulations. [...] can these folks legally receive bitcoins? Or, is that considered Pay and Business?

Well, the IRS thinks that Bitcoins are money. Various state-level tax agencies think that Bitcoins are money. I'd guess that Bitcoins received for legal advice would probably be considered 'pay.'
As for dispensing medical advice...

Van Thiel claimed he performed abortions, removed sebaceous cysts, treated sexually transmitted and life-threatening diseases and provided ozone treatments at “unbeatable prices” in exchange for Bitcoins, gold and silver and firearms.

There has been someone who was prosecuted for unlicensed practice of medicine in exchange for Bitcoin.