[Yusup Hizirov (OP)]

What phase of the moon coincides with perihelion.

No correlation at all.  Perihelion (when Earth is closest to the sun, in early January) can occur at any moon phase, depending on where the moon is at the time. This year, perihelion occurred with about a half moon.
Similarly, perigee (when the moon is closest to Earth) can also occur at any moon phase, depending on where the sun is at the time. Next full moon is March 9, and perigee the day after, but in July, perigee happens again with a half moon.

Does the ellipse of the moon and the sun rotate, if so, at what speed?

[Halc]

Does the ellipse of the moon and the sun rotate, if so, at what speed?

It's called apsidal precession, and the moon does it once every 106 months (not quite 9 years) or so.
Earth does it as well, about every 112,000 years.
The motion of the sun is fairly erratic and is not approximated as an ellipse.  The center of gravity of the solar system is within the sun about half the time.

[Yusup Hizirov (OP)]

This means that the orbit of the moon today is extended along the orbit of the earth.

Then two years later, the moon’s orbit will be stretched perpendicular to the Earth’s orbit.
https://images.app.goo.gl/jCVZeeHcyqEKYnQ69

[Bored chemist]

I still want to know what this "mystery" is.
Even if we couldn't calculate the Moon's orbit to great precision (and we can), then we could just watch  the dratted thing and see what it does.

[Yusup Hizirov (OP)]

I still want to know what this "mystery" is.

It is believed that at perihelion, the moon is at 50 thousand km. closer to Earth than aphelion, why this difference is not visually observed.

[Halc]

It is believed that at perihelion, the moon is at 50 thousand km. closer to Earth than aphelion, why this difference is not visually observed.

Again, the distance from Earth to moon at perihelion and aphelion is pretty random, and might be the same on a given year.  I think you mean perigee and apogee, where the difference is about 43000 km.

It is very visually observable.
https://upload.wikimedia.org/wikipedia/commons/c/c8/Lunar_perigee_apogee.png
This shows why a solar eclipse traces a wider swath if near perigee than one near apogee.

[evan_au]

the inner core of the earth rotates faster than the surface of the earth. The core of the earth is dragging the surface via viscoelastic friction between the core and mantle.

The theory of "Super-Rotation of the core" is still controversial, with some early claims as high as 3°/year.
- Current measurements show it is < 0.1°/year
- Some theories suggest it is < 1°/million years
See: https://en.wikipedia.org/wiki/Inner_core_super-rotation

Measurements of super-rotation on Earth are limited by the number of seismic monitoring stations 50 years ago.

Measurements on the Moon are more limited, given the small number of seismic monitoring stations placed on the Moon 50 years ago. It is thought that the Moon has a solid iron inner core, surrounded by a semi-liquid outer core.
See: https://en.wikipedia.org/wiki/Internal_structure_of_the_Moon#Core

Measurements on Mars are even more limited by the single seismic monitoring station placed on Mars last year.

[Yusup Hizirov (OP)]

What force rotates the orbit of the moon.

[Bored chemist]

why this difference is not visually observed.

Why do you think it is not observed?
Perhaps the most wide scale observation happens when we get an annular eclipse rather than a total eclipse.

https://www.timeanddate.com/eclipse/solar/2020-june-21

Millions of people will see it.