My question is this, if you play a really great round as your first ranked round will your average stay fixed for the entire rank or does bad rounds affect the average negatively?
An average is the average of different values, so each round will influence the display.
I am at Master now, do I have to keep playing 63 or better rounds in order to advance or will I advance regardless of subsequent round scores. I had a couple of bad rounds in a row, so I wanted to know if I have to play two good rounds now to fix the damage.
Play your scores, good or bad, and - when the time comes - you will be promoted.
You need a minimum of 40 scores (for "saturation") and an average of 61.00 or better. Your 40 best scores will be used for calculations, the others - the bad ones - will drop out.
Don't fixate on the idea to compensate bad rounds with good ones. You have to collect 40 rounds which - together - result in an average of 61.00.
Beware that there are other mechanisms which may accelerate your promotion!
Is there anywhere I can see how many ranked rounds are still required to reach Tour Master?
There is no fixed number - "saturation" is important which you may count in your score history or experience it - when the average drops mightily!