You might know that all matter is made up out of atoms. Now, atoms themselves have a core, or nucleus, and electrons orbiting around the nucleus. The core has positive charge, the electrons have negative charge.
When you are rubbing the glass rod with the silk cloth, electrons are stripped away from the atoms in the glass and transferred to the silk cloth. This leaves the glass rod with more positive than negative charge, so you get a net positive charge.
Why do the electrons go from glass to silk and not from silk to glass? That depends a lot on the minute details of the material. Ultimately, for every two materials there is one of them where electrons are happier. It just turns out that for glass and silk, electrons are happier at the silk cloth.
Now to your second question. Here, the important thing to note is that in your typical solid material, the positive charges, which are the cores of the atoms, cannot move around much. They are locked into a rigid structure. The tiny electrons, however, can move around much better. That's why the glass rod can induce a net negative charge at one end of the paper clips.
EDIT: Let me add that there should also be some attraction between the silk and a bunch of paper: The electrons in the paper will be pushed away by the electrons in the silk, leaving the end of the paper that is closer to the silk with a net positive charge that then gets attracted. However, it might very well be that in your silk cloth the electrons are overall too spread out to have a strong enough attractive effect.
Answer to your second question:
If you take any point inside the conductor, there is a non-zero potential at that point due to the external field, and as you correctly said, the potential of the sphere should be zero as it is grounded. So a net charge (+ or - depending on the field) will accumulate on the conductor to bring the potential inside the conductor to zero. That's why those negative charges don't flow to the ground.
I'm not exactly sure about the answer to the first question.
Best Answer
The earth is at a much larger electrostatic-potential, than the sphere is. So, positive charges flow from the earth to the sphere, the number (I speak of number because charge is quantized) of which is equal to the number of electrons that accumulate in the farther end of the sphere. So, due to balance of charges at the farther end, the sphere loses its negative charge on the farther-end, and the positive charges get distributed uniformly all over the sphere. This explanation is based on the conventional flow of current.
Actual scenario is that, not positive charges, but rather the free-electrons in the metal of which the sphere is composed, are the real charge-carriers. These electrons, because of the existing electrostatic potential-gradient, existing between the earth and the sphere, flow through the conducting-wire, from the sphere, to the earth. Then the positive-charges are distributed uniformly, all over the sphere.
Satisfied yet?