The truth is much more complex than that. The hierarchy of motions you cited are not independent of each other. They influence each other, from large to small and are interconnected.
Newton's law applies relative to fixed, non accelerating coordinates only. In any rotating coordinate system, such as earth coordinates, corrections must be applied for the rotation of our planet. For the long-term movements of objects these corrections can be significant - even dominant. Even with this there are, as you suggest, additional factors due to the rotation of the earth around the sun, the rotation of the solar system around the center of our galaxy, and again of our galaxy around the gallactic cluster in which it resides. The relationships among these is not precisely fractical, but are nested as you noted. Happily, owing to the distances involved (1) from the sun to the earth (2) from the center of the galaxy to the sun, and (3) from the unknown center of our gallactic cluster to the center of our galaxy, etc. the angular velocities are vanishingly small, and it is sufficient for most human needs to consider only the effects of our rotating earth.
With this in mind Newton's law becomes Force = mass x acceleration, or
Force = mass x ( apparent acceleration in local coordinates - centrifugal force - Coriolis Force) where these fictitious forces are simply corrections resulting from the fact that earth coordinates are not fixed, but rotating with the earth.
centrifugal force can be thought of as the outward force on an object resulting from the earth's rotation. It is zero at the poles and a maximum at the equator. It is directed perpindicular to the earth's axis and therefore has components in the local vertical and in the northern hemisphere in a southerly direction.
Coriolis "force" is a bit more complex. Named after the French hydrodynamicist who first wrote about it, it is a correction associated with a body that is moving relative to the local (rotating) earth coordinates. It's magnitude is proportional to the local velocity of the object and to the angle between the direction of movement and the axis of the earth. Its effect is to turn moving objects to the right in the Northern Hemisphere and the left in the Southern. The atmospheric flow around low pressure areas is, as a result, counterclockwise in the northern hemisphers and the opposite in the South (same in your sink). Coriolis also affects oceran currents and is the reason we have westerly trade winds near the equator and westerlies at mid latitudes.