A merry-go-round moves in a circle at a constant speed. Is the merry-go-round accelerating?

No, since the merry-go-round's speed is constant, it's speed is not changing; hereby, acceleration is impossible; for, by definition, acceleration must involve some change in speed. Without a change in speed, an object cannot accelerate. Acceleration accounts for some change in speed over a certain amount of time, if the change of speed is zero, the acceleration must necessarily be zero.