Do solar flares move the dust on the surface of the Moon? The TV show For All Mankind has a reputation for scientific accuracy, so I think it's fair game to ask if it lives up to this reputation.
In this video clip, we see the dust on the surface of the Moon being perturbed by radiation coming from the Sun during a flare:
https://www.youtube.com/watch?v=YzlP403rUGA

Does this actually happen in reality?
 A: I did some digging around and I found that solar storms can quite significantly erode the surface of the moon. From NASA's website, they say:

"We found that when this massive cloud of plasma strikes the moon, it acts like a sandblaster and easily removes volatile material from the surface," said William Farrell, DREAM team lead at NASA Goddard. "The model predicts 100 to 200 tons of lunar material – the equivalent of 10 dump truck loads – could be stripped off the lunar surface during the typical 2-day passage of a CME."

I also did a simple calculation to test how significant such an effect could be:
The luminosity of the sun is typically known to be $L_{\odot}$. A solar flare could increase its luminosity by $\Delta L$ such that the total luminosity from a solar storm is on average $L_{\odot} + \Delta L$ which will be distributed on a sphere of radius $R$, the distance between the Moon and Sun. For every photon, $p = E/c$ which means $P = I/c$ where $P$ is pressure and $I$ is energy density per unit time. Hence, the pressure on a dust particle on the moon is
$$P = \frac{I}{c} = \frac{L_{\odot} + \Delta L}{4\pi R^2 c}.$$
Consider a dust particle of density $\rho$ and radius $r$. The force from radiation on this particle will hence be
$$F_{\text{radiation}} = PA = \frac{(L_{\odot} + \Delta L) \pi r^2}{4\pi R^2 c}$$
The gravitational attraction between the dust particle and moon is given by Newton's law:
$$F_{\text{gravitation}} = \frac{Gm_{\text{moon}} (\frac{4}{3}\pi r^3 \rho)}{r_{\text{moon}}^2}.$$
So, let's create a dimensionless parameter $\alpha$:
$$\alpha = \frac{F_{\text{radiation}}}{F_{\text{gravitation}}} = \frac{(L_{\odot} + \Delta L) r_{\text{moon}}^2}{16\pi G \rho c R^2 r m_{\text{moon}}}$$
If $\alpha$ is around $1$, we can say the forces are within comparable magnitude.
Plugging in values when taking $\Delta L = 0$:
$$\begin{align*}
L_{\odot} &= 3.8\times 10^{26}\;\mathrm{W} \\
r_{\text{moon}} &= 1737400\;\mathrm{m} \\
G &= 6.67 \times 10^{-11}\;\mathrm{m^3 kg s^{-2}} \\
\rho &= 3340 \;\mathrm{kg/m^3} \\
c &= 3\times 10^8 \;\mathrm{m/s} \\
R &= 1.52\times 10^{11}\;\mathrm{m} \\
r &= 10^{-6}\;\mathrm{m}
\end{align*}$$
gives us the value of $\alpha \approx 0.0067$. Depending on the value of $\Delta L$, the effects of solar radiation could be even greater.
In conclusion, the effects of solar radiation are not negligible, but probably more exaggerated in For All Mankind.
A: Yes, why not? NASA studies found that that the moon do gets periodically "sandblasted" by intense solar storms that can strip tons of material from the lunar surface. Charged particles from solar flare/winds collide and eject material on the moon's surface. This process is named sputtering.
A new computer simulation was done and found out that this sandblasting effect kicks into high gear during intense bursts of solar plasma charged gas known as coronal mass ejections (CME). As CME contains heavier ions like helium, oxygen, and iron, they are more likely to collide with much greater force and eject atoms off the surface. A strong CME can hurl about a billion tons of solar particles at up to a million miles (1.6 million kilometers) an hour in a cloud.
Reference

*

*Solar Storms Are "Sandblasting" the Moon, NASA Study Hints - National Geographic article

*NASA Study Finds Solar Storms Could Spark Soils at Moon's Poles - NASA article

*Lunar dusty plasma: A result of interaction of the
solar wind flux and ultraviolet radiation with the
lunar surface by E A Lisin et al 2015 J. Phys.: Conf. Ser. 653 012139 (PDF)

A: The show probably exaggerates but solar wind and solar flares do have effects.

*

*A sputtering effect where the energetic particles  knock loose atoms from the material. This can apparently remove several tons of material during a solar storm.


*Accumulation of charge that can lead to sparking and very localize melting of the dust. This modifies the structure of the material where the sparking occurs.
A: Yes lunar dust does levitate above the Moon's surface because of electrostatic charging due to the solar wind:

*

*On the day side it extracts electrons from the dust making it positive.

*On the night side of the moon, the solar wind charges the dust negatively.

So on both sides, dust levitates due to electrostatic repulsion.
This is the "Moon fountain" effect anticipated in science fiction book by Hal Clement "Dust Rag" published in Astounding Science Fiction, and really observed by several NASA Apollo missions later on.
Please see document from NASA archive site.
A: 
Does this actually happen in reality?

No. Or at least not unless the solar flare was an extinction-level event. It is just added for dramatic effect.
