I'm wondering what will happen if the core of the Earth became cold? Will it affect the rotation of the Earth? or will it change the magnetosphere?
The movement of the molten core generates the magnetosphere, as your posting mentions. Here is a nice succinct article about how the magnetosphere protects earth.
CONSERVING EARTH'S ATMOSPHERE Earth’s magnetosphere is also vital in preventing our atmosphere from being pushed out into space by the pressure of the solar wind.
BIOLOGICAL SHIELDING If not for the magnetosphere pushing away electrons and protons of the solar wind, the charged particles would inflict doses of harmful radiation to life on Earth.
I had in my mind that Mars' lack of a magnetic field (presumably because of frozen solid core) is the reason it lost its atmosphere - swept away by the solar wind. The linked article (I may have to look at this painful ad-heavy sciencing.com site some more...) notes that during a recent burst of solar wind enveloping both planets, Mars lost 10 times the atmosphere Earth did.
Frozen core also means no plate tectonics and possibly less volcanism: both positive things for humans I think. Not a good enough reason to freeze the core, though. In case you were considering that.
It should not affect the rotation of the Earth but it definitely would have an effect on the magnetosphere, volcanic activity and plate tectonics. The magnetosphere would likely be reduced by some 95%, the iron in the core left behind would still create a small magnetosphere but with the dynamo gone we'd lose the vast majority of it. Volcanic activity would also drop, assuming the core goes cold, the mantle would eventually follow too. (Unless it cooled at the same time for the same reasons.) Plate tectonic would eventually stop too.