Does the moon produce a measurable tidal-effect on the Earth's (liquid) core? If so, how strong is it? Would it play a factor in other geological effects like earthquakes, volcanoes, etc?
The gravitational fields of the Sun and the Moon do produce measurable effects on the shape of the Earth. The tidal distortion of the solid Earth (and the liquid outer core) is often referred to as the Earth Tide to distinguish it from ocean tides.
The gravitational effects of the Earth tide do cause small strains in the Earth which might influence geophysical processes such as volcanic eruptions and earthquakes - however, it is hard to prove specific cause and effect instances.
Agnew, D. C. (2007). Earth Tides, pp. 163-195 in Treatise on Geophysics: Geodesy, T. A. Herring, ed., Elsevier, New York.
According to https://www.lhup.edu/~dsimanek/scenario/tides.htm, the earth crust moves about 20 cm in response to tidal forces. I expect that if stress is built up in a fault line, this would play a part in releasing it, but this is not my area of expertise.