Mercury has a 58.646 day rotation period. What is the speed at which a point on the planet's equator moves due?

a) Mercury has equatorial radius of 2439 km.
So the total distance travelled for 1 rotation
 = 2TT(2439)km
 = 15324.68896 km
Speed = distance / time
 = 15324.68896 km / 58.646 days
 = 261.308 km per day
 = 10.8878 kmph
 = v
Earth's speed is about 1600 kmph in contrast
b) v/c = 1.0088318 X 10^-8 (10 part per billion with 0.88% error`)
1+z = ?[(1 + v/c)/(1 - v/c)]
 = ?[1.00000002]
 = 1.00000001
{to be exact it is
= 1.000000010088318 / 0.999999999999999898225386}
z = 0.00000001 = 1 X 10^-8
Shift in wavelength would be
12.5 X (1 X 10^-8) cm = 1/8 micron.
Comment : You can't detect it.
Since you are working in 12.5 cm wavelength in Microwave region correspong to 2,398,430 Hz, a shift in frequency of order
2398430 X 10^-8 = 0.024 Hz.
I am not aware of someone giving you a frequency descriminator of almost 1/40 Hz. Less than 100 Hz is difficult to detect in frequency discriminator. In the reading of frequency meter you wont get numbers running to 0.001 Hz, everchanging like noise. I haven't come across even people working in sub-Hz part of spectrum.
It looks like you need to use it to beat with a 400 Hz (ultra coherent) signal and note the deviation on a pen-recorder plot on paper, as the beat at 1/40 is very sub-audible, if you need to establish the frequency.
For such studies you need frequency sources of coherence that can offer better (stringent) deviation in frequency than 1 part per billion.
If you start off with Green light (in stead of 12.5 cm) at 0.5 micron(599.585 THz) so that you can sense the frequncy deviation ('z' above) in kHz range, to a figure 600 kHz (0.6 MHz) radio frequency. So the project should be to get Green light from Mercury surface, after extensive down-converting (that is the term used in 'radio technology'; but I never heard any one down-converting by 6 or 7 stage mixing by a factor of a billion) extract this information.          {I wouldn't do that}