Effect on pulsar signals of Earth's orbit Do pulsar signals 'change' with our orbit?
For example, if a pulsar was in Sagittarius, our orbit would have us 300 million km closer to it each May than we were in November.
And also see us moving towards the pulsar at ~30 km/s in February, and moving away from it at the same speed in August.
Part 1:
Can we see the effect on a pulsar's signal of the extra 300 million km the pulse must travel to reach us in November compared with in May? 
I know that pulsar signals are extremely regular ("better than an atomic clock"), so I guess an extra 16 minutes gained over 6 months, and then lost over the next 6 months must be measurable?
(Ole Römer and Christiaan Huygens pretty much managed it in the 1670s, after all.)
Part 2: If our pulsar emits a pulse in RF, would we be able to measure the blueshift in that frequency in February (and then the redshift in August) in that short signal? 
Are pulsar signals 'coherent' or does their pulse cover a range of frequencies? If a range, can we measure this doppler shift in the range of frequencies? 
As I wrote this, I realised that Part 2 must apply to the light from any star, quasar, FRB, or alien broadcast. So, as above, but not just to pulsars: Is 30km/s fast enough for us to measure the resulting doppler shift in light from stars/pulsars/whatever?
Part 3: How does our Solar System's even faster (~200km/s) movement around the galaxy effect EM signals? Is light from "ahead of us" or "behind us" measurably shifted? Or, do we assume it is, but can't measure the shift as we're not changing direction?
In 250 million years' time, should we see a change in frequencies from sources outside the galaxy (thinking quasars).
(Apologies in advance if I have any of my constellations/months wrong!)
 A: 
Part 1: Can we see the effect on a pulsar's signal of the extra 300 million km the pulse must travel to reach us in November compared with in May?

For dispersion, pulse profiles, etc, the added distance has no observable effect (to my knowledge).  It is completely negligible compared to the typical distance between us and the pulsar (typically kpc, $\sim 10^6$ times farther away).  The additional light travel time is observable.

Part 2: If our pulsar emits a pulse in RF, would we be able to measure the blueshift in that frequency in February (and then the redshift in August) in that short signal?
Part 3: How does our Solar System's even faster (~200km/s) movement around the galaxy effect EM signals?

Yes, absolutely to both parts.  At any given time we measure the radial velocity (velocity along the line of sight) to the pulsar.  Over the course of a year, a periodic component due to the earth's motion in the solar system is clearly observable.  For this reason, pulsar timings are usually converted to 'barycentric' coordinates---the center of mass of the solar system, to remove the earth's motion.  As you guess, because the solar system's motion is very nearly constant, changes from that (i.e. orbiting the galaxy are completely negligible).
Whether different types of relative motion are observable in general astronomical observations depends on the type of source (i.e. if the signal is narrow-band) and the quality of the observations (signal to noise and the spectral resolution).  Any source with a narrow enough line (i.e. of comparable or smaller width than what you're trying to observe, e.g. $\sim 30$ km/s) should work.
A: The answer to all of these questions is yes. Pulsars have such regular, clock-like pulses that their "ticks" can be used to measure all of these phenomena in exquisite detail. One of the things you have to realise is that measuring a pulsar's frequency is not a single measurement of frequency. If there is a frequency change or drift you get a cumulative effect on the pulse arrival times that may build up with time that then makes even very small effects observable. This is how pular timing networks are now being set up that may be capable of detecting low frequency gravitational wave signals as they propagate past the Earth.
To get to your questions.


*

*Yes of course. The difference in pulse arrival times is massive - as large as 16 minutes. This combined with the Earth's orbital speed of 30 km/s that modulates on the yearly cycle puts a comparatively enormous signal into the pulsar pulse arrival time sequence. In fact the delay in the pulse arival time received at the Earth is the sum of the Roemer delay (which is just the light travel time across the Earth's orbit), the time-dilation caused by the relative motion of the Earth and pulsar, the time-dilation caused by the presence of your observatory within the potential well of the Earth and the solar system and the "Shapiro delay", which is the delay caused by the signal from the pulsar travelling through the curved space-time associated with the Sun and planets. All of these things have measurable effects on the pulsar timing - e.g. see here.

*Yes of course. $\pm 30$ km/s is a massive doppler shift compared with the limits of sensitivity. For pulsars you are looking at patterns in the time of arrival of the pulses, not Doppler shifts in what is essentially a fairly featureless continuum. For other astronomical objects, the methodology varies widely, but for instance exoplanet searches regularly look for velocity changes of 1m/s in stellar spectra.

*Yes, the relative motion of the Sun with respect to the pulsar will cause a doppler shift and time-dilation. Because the timescale on which this changes is so long (millions of years) then this is essentially a zero point against which all the shorter timescale variability is measured.
The orbital period of the Sun around the Galaxy is 200 million years or so and it orbits with a speed of 220 km/s. So, depending on where the pulsar/quasars/whatever is, then the frequency of the signals we receive will be modulated on this amplitude and period.
Changes in pulsar frequencies can be measured in the best cases to about 1 part in $10^{15}$, but this rests upon a simple model for how pulsars spin down with time. These timescales are short - thousands of years - so disentangling the evolution of the spin down rate from secular changes due to the long-term motion of the Sun around the Galaxy would be very difficult/impossible on human timescales.
Quasars present a different problem. They are long lived and emit (presumably) at constant rest frequencies. Therefore we would expect those frequencies to gradually change as the Sun orbited the Galaxy. The question is whether that can be measure on human timescales. Over a decade, the velocity of a quasar might change by 0.02 m/s and that is simply not detectable in the spectrum of a quasar (or any other astronomical object) at present. However, if you wait tens of millions of years (or probably even thousands of years) then the signature of the Sun's motion around the Galaxy would be easy to measure.
