Does the moons gravitational pull have an effect on the Earths molten Iron core?

certainly

all mass has gravitational interaction depending on M and 1/R^2

the laws of orbital motion also apply

It is the Center of Gravity point that follows the idea orbit.

However the core is closer to the CG and the tidal effect would be smaller

interestingly there is NO total gravity at the center of the earth all the mass in concentric shells cancels out. Newton had to prove that to make his calculations work

since the earth mass "causes the gravity that affects the moon, the mass of the moon "causes the gravity of moon on earth, and all parts of the earth

Yes. The Moon and the Earth orbit their mutual center of gravity (barycenter). The Earth's liquid core has a "tide" also but is very much smaller than that of the oceans.

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