As a holder of an ARCT in piano performance and professional musician, I find this topic to be interesting.
Of course I have a slight bias, but I believe an ARCT is a great background. To the original poster, I can relate to your questions because I once pondered them myself.
Professional means being payed real money for ones playing, teaching, or writing music. If one has an ARCT or is working towards one, they can be rest assured (good social skills will help) they are prepared to teach at least beginner to intermediate level students.
An ARCT is quite advanced, though again, relative is the term to keep in mind. Most concert pianists, ie ones with large repertoires of concertos and solo programs, would see the associate's diploma as a basic starting point.
To the general public and the vast majority of musicians out there, an ARCT places one with an extremely high level of technical skills and musical training. It really is a highly respectable degree of musical training one can be proud of throughout life.
My advice is to think about what you want to achieve in music. I took 2 years after high school to work and complete my ARCT. I did not pass my first try and achieved it at age 20. By then, I was quite confident in my classical piano abilities.
It is my opinion that an ARCT, with the combination of a musical curiosity and imagination, gives one the foundation and ability to play anything they desire on the piano.