Arbitrarily discarding/cancelling Radians units when plugging angular speed into linear speed formula? Why is the radians implicitly cancelled?  Somehow, the feet just trumps the numerator unit.
For all other cases, you need to introduce the unit conversion fraction, and cancel explicitly.  Is it because radians and angles have no relevance to linear speed (v), so they are simply discarded?
Radians are dimensionless, i.e. there is nothing to cancel. This is because a radian is defined as a length over a length:
$$\frac{\text{radius}}{\text{circumference}}$$
So a radian is defined as

$1~\mathrm{radian} = \frac{\mathrm{radius}}{\mathrm{circumference}}$?

No, the radius over the circumference is a pure number equal to $1/(2\pi)$.

"One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle" is the more accurate definition. [1]

[1] http://en.wikipedia.org/wiki/Radian