Learn English – a more professional term for the ‘back-of-the-envelope’ calculation

"Back and forth" is the more correct idiom, because, well, that's the idiom. There's nothing to stop you from saying "forth and back" — a little voice is repeating the subtitle to The Hobbit, which is There and Back Again, to me — but you won't be using the English idiom, you'll merely be speaking words.

The figurative sense of professional that you describe is perfectly fine, as long as you're careful how you use it--for example, you can describe yourselves as "projecting a professional appearance."

If that's not a satisfactory answer, consider using different words in different contexts: you have a refined appearance, give a polished performance, are classy in your interactions with the public, and responsible in your dealings with booking agents.