1.The object is attracted to the center of gravity of the earth, attracted to the center of the earth. The gravity on the object is unaffected by the spinning of the earth. The object does not in any way “know” the earth is spinning or what direction etc. So if it is held far above the earth and let go, then it will fall straight toward the center of the earth and the earth can spin under it.
If instead of just letting it go, you take it to some elevation and throw it, then it will still be in free fall - that’s because no other force is acting on it. It will also still be attracted to the center of the earth, but will travel in an arc. It still only cares about the center of the earth and which way you threw it though. You could throw it to move along with the spin of the earth or the opposite way, or perpendicular, etc.
2.Poles don’t matter for gravity. In fact the earth doesn’t even have to be round; it could be a cube. The object will still always be attracted to the center of the earth. If the earth isn’t round, then elevation alone won’t tell you how far it is to the center. For example whenever we are above point A on the earth, then a mile above the surface means 1,000 miles from the center of the earth. But above point B, a mile above means 1,005 miles from the center (if and only if not round).
Finally, it doesn’t usually come into play, but the distance to the center tells you how strong gravity is. Further from center and gravity gets weaker. This doesn’t usually matter because even if we are thousands of feet above the surface, we are still around the same distance (thousands of miles) from the center.