The screw wouldn't be weightless. Imagine that it's a metal toothpick instead of a screw, that is being attracted towards to a large flat magnet. If the toothpick is 'falling' horizontally in relation to the surface of the magnet then each side of the toothpick is attracted with equal intensity towards the magnet. If the toothpick were weightless then it would be able wander around into different positions as it falls, and could even rotate into a diagonal or vertical position.
And if for some reason you think that the vertical position is most ideal for a metal body that is attracted a magnet, perhaps with the magnetic properties more concentrated when falling as a spear, then you have the opposite problem and the metal body is still not actually weightless and able to rotate into any position.
It is obvious that falling metallic bodies wouldn't experience weightlessness when attracted towards a magnet. By extension, weightlessness would not occur with any pulling phenomenon.
The depths of your misunderstanding of very basic Newtonian physics is so profound that I think it's probably best if you work on that before you dip your toes in the wonderful world of relativity, as you have in some of your other posts here.
The sensation, or state, of weightlessness occurs when there is no force acting between the component parts of a body. For a human, we 'feel' weightless when there is no tension or compression in our body - witness the floating hair of the girl in your vomit comet photo. This can either be because there is no force acting on us at all - a pretty much impossible situation - or because all of the particles that we are made of are accelerating at the same rate due to some external force. If we accelerate due to some external influence, like going up in a lift, we feel the acceleration because the lift only applies a force to our feet. Our feet have to accelerate our ankles, our ankles push up through our legs, hips, etc - there is compression in our bodies due to this transmitted force. But if we fall due to gravity, we feel weightless because all of the particles in our bodies experience a force proportionate to their mass, meaning there is no tension or compression.
You can see this in your balloon diagram.
At rest, a balloon filled with water will sag down, as the elastic material stretches to counter the weight of the contents. In freefall, the water and the balloon are all being acted on by gravity in proportion to their respective masses, so the tension in the balloon returns it to a natural sphere.
There's nothing in any of this that contradicts the equivalence principle, or that falsifies our typical, newtonian understanding of gravity.
A common misconception regarding spaceflight is that the weightlessness experienced in orbit is due to there not being any gravity. This is completely wrong - there is only a small reduction in g in low earth orbit. The reason astronauts feel and appear weightless is because they are perpetually 'falling' (ie accelerating) towards the centre of the earth, but are travelling so fast horizontally that they trace a perfect circle - the orbit - around the earth. If you could travel fast enough and overcome the drag from the atmosphere - I believe the figure is around 17,600mph - you would experience weightlessness at the earth's surface. If you were flying an aircraft at 17,6000mph you would need to 'push' to 0g just to maintain the same altitude. Oddly enough, if aircraft were equipped with extremely accurate g-meters, in straight and level flight the accelerometer wouldn't actually say 1g, it would actually be very slightly less than this, reducing further as speed increases.
As an aside, accelerometers are something of a confusion - the standard is to read 1g at rest on the earth's surface, and therefore 0g in freefall. This is very confusing, as of course freefall is acceleration, but there is some logic to it.