Does the vector representing the angular velocity of a wheel rotating about a fixed axis necessarily have to lie along that axis? Could it be pictured as merely parallel to the axis, but located anywhere? Recall that we are free to slide a displacement vector along its own direction or translate it sideways without changing its value.

It is necessary that the vector representing angular velocity of a wheel rotating about a fixed axis have to lie along that axis because the direction of the vector depicts the nature of rotation of wheel, that is, if the wheel is rotating clockwise around the axis or counterclockwise.
The vector will be parallel to the axis of rotation but the direction of the angular velocity vector will indicate if the wheel is rotating clockwise or counterclockwise relative to the axis of rotation.
In case the axis is fixed, one can choose to place the angular velocity anywhere in the region but the vector must be parallel to the axis of rotation and pointing in the direction corresponding to its direction of rotation.
However if the axis of rotation can change, locating the angular velocity vector anywhere could be misleading, and non-conclusive.
For example consider a situation in which the object rotates counterclockwise about its axis of rotation and revolves around another axis in the same direction. In this case if the angular velocity is placed anywhere in the region, one cannot conclude if the vector highlights the direction of revolution of object or the direction of rotation.