If an asteroid twice the size of Earth passed super close would half of the Earth be pulled towards it? So let's say that an asteroid the twice the size of Earth passed by going relatively slow say in between the Moon and Earth.

Among so many other things (and perhaps you can touch on some of the biggest 'other' things), would anything on the asteroid's side of Earth be essentially 'sucked' towards the asteroid. That is if I am standing outside and it passes would I lift off the ground and float towards it? 

Conversely, would things on the opposite side experience two times the standard weight? That is if I am standing on the other side, would I feel heavier as the asteroid passes?

I realise there would be many other devastating effects but I'm most interested in this one...

The possible answer I suspect is that objects on Earth will not 'feel' anything as the Earth as a whole will move towards the asteroid and not every little thing inside Earth...
In a close approach, Earth would feel tidal forces: the near side to the approach would be affected by the vistor's gravity more than the far side, because the force of gravity decreases with distance from a mass. If you were standing on the far side, though, your effective weight wouldn't increase -- it would decrease! This is because the visitor would be pulling the center of the Earth more strongly than it would be pulling on you, again because of the gravitational falloff with distance.

In a very close approach, closer than the Roche limit, the tidal forces would overcome the Earth's gravity completely, and Earth would begin to break apart. So it matters very much how close an approach we're talking about. Assuming the visitor has the same density as the Earth, the critical distance is about 10 Earth radii, or 64000km -- about a sixth of the distance from the Earth to the moon.