Earth's gravitational pull is exerted from the centre of its mass, so all objects are pulled in what we perceive to be a 'downwards' direction towards it. If it were possible for you to be at the exact centre of Earth's core, you wouldn't experience its gravitational attraction... only the weaker gravitational pull from the Sun and other distant large celestial bodies.
Although the Earth is spinning, you and everything on it including your house, the atmosphere and the oceans are spinning with it at the same speed. Since the speed of rotation is almost constant, you don't experience any sensation of movement because everything else is moving with you, but you would notice if it suddenly slowed down (or speeded up). For the same reasons, you don't experience the forward motion in a speeding car at constant speed, but do notice if it suddenly accelerates or brakes.
The flat static Earth picture and my comment about the spin-dryer effect was intended as a joke. The evidence for Earth being a sphere (albeit slightly flattened at the poles because of its spin creating a bulge at the equator) is overwhelming to all but the most stubborn die-hard believers in a flat Earth.