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!)