The working distance of an objective lens is the distance between the front lens mount and the focal plane. Lenses with high NAs usually have a lower working distance.
The Depth of Field of a lens (microscope objective in this case) is the distance above and below the plane of focus that is acceptably "in-focus" to the eye. It usually decreases as the objective NA increases.
Properties of Objective lenses
As described by Snell's law, refraction of light is a function of the change in refractive index of media through which the light travels. When light passes from a microscope coverglass (CG) into some medium between the CG and objective ("immersion medium"), refraction can occur. If, for example, the "medium" between the CG and objective is air (n=1.000), Light rays impinging at a wide angle (e.g., b') will be refracted outside of the collecting angle of the lens. At a greater angle the light will be reflected (see Critical Angle).

When the refractive index of the immersion medium approaches that of glass (n=1.515), then light rays will not be refracted, and the lens will collect more light. The medium in this case is Immersion Oil.

Remember that Numerical Aperture, or the measure of the light gathering power of a lens, is a function of the refractive index of the immersion medium. Using the wrong immersion medium will greatly decrease the Effective NA of a lens.