Does Doppler shift affect apparent pulsar frequency? We know that redshift and blueshift is the result of the frequency of light waves reflected or emitted by an object lengthening or shortening due to the relative velocity of the observer moving away or toward the object. This is the Doppler effect.

However, it was stated in response to a question I asked on a sister site that the Doppler effect would also result in the apparent frequency of a pulsar's pulses being longer or shorter due to the relative velocity of the star. Given that the speed of light is constant regardless of the frame of reference (a concept I have a really hard time understanding by the way), is this correct?

My thinking is that it is incorrect. Using the analogy of a launcher lobbing medicine balls at me at a fixed rate, I can "see" the rate of launch by the frequency of when I get hit. However, if the launcher is moving away from me, the velocity of the balls lobbed at me will be slower (relative to my frame of reference), and thus will hit me with less frequency. However, if the launcher were to adjust its firing speed (i.e. speed of the ball relative to itself; frequency remains constant in this analogy) so that the balls always hit me at the same speed (thereby simulating the concept of the always-fixed speed of light), the time between them would not change.

So, given the always-constant speed of light, is it correct that a pulsar's pulse rate would appear to be different at different relative velocities?
Yes, it is correct.

Consider the classic example of a (say) $1000$ Hz siren on an ambulance approaching you.  Once a single wave is produced, the source moves towards you before the next wave is produced.  The wavelength of the sound waves in the air is reduced, so the frequency you hear is increased.  Simple, basic Doppler shift...

But consider the case where the siren is being switched on and off with a five-second overall period.  So you have a $1$ kilohertz signal modulated at $0.2$ Hertz. 

But the same argument applies to this modulation frequency.  A chain of $1$ kilohertz waves is generated for $2.5$ seconds and sent on its way.  The ambulance approaches you for the next silent $2.5$ seconds, then starts the next chain on its way, closer to the last chain because of the distance travelled in the silent phase.  The same Doppler shift factor applies to the carrier frequency and the modulation frequency.

The same physics almost derailed the Cassini-Huygens probe. The extreme velocity changes of the probe caused a Doppler shift in both the transmission frequency and the data rate carried by this transmission. The first was handled;  the second was not!

See http://www.thespacereview.com/article/306/1

or

http://spectrum.ieee.org/aerospace/space-flight/titan-calling