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.
Charged objects have an electric field in their vicinity. The air will always contain a small number of ionized particles - this can be a result of cosmic radiation, local electrical activity, or just the chemistry of molecules. Now if your comb is positively charged, negatively charged ions will be attracted to it and positive ones will be repelled. Over time, this results in a neutralization of the charge on the comb.
The question then becomes - what is the major contributor to the conductivity of the air? A 1988 research report by Hugh R Carlon, Electrical properties of atmospheric moist air: a systematic, experimental study had the following in its abstract:
All results are consistent with the view that ions are produced in moist air [...]
Reproducing figure 3 from that paper:
you can see that they measured a charge carrier density that is logarithmically dependent on relative humidity - and charge carrier density scales with conductivity (all other things being equal).
This is the reason, incidentally, that most ESD (electrostatic discharge) damage occurs in winter: when the air outside is cold, the relative humidity inside (where it is warm) will be low; thus the conductivity is low, and objects have a chance to build up charge without it leaking away so quickly.
One more figure from the same paper, showing the discharge current measured for different voltages, at different levels of humidity. Once again, it is hard to escape the conclusion that ionization of moisture in the air is responsible for the conductivity (and thus for the discharging of your statically charged comb):
Incidentally, in a dusty environment (see @Boris's answer) you may observe neutral dust particles being attracted to the comb (because of polarization of the particle) - but once it touches, it either sticks, or it will "jump away", having acquired a small amount of charge and therefore being now repelled. In so doing, these particles can be responsible for some charge leakage. But in clean air, moisture is the key component of conductivity.
Best Answer
Air, especially if humid (high water content), can conduct electricity to some extent.
Electrons are being repelled from the negatively charged comb and some will gradually go into the air and move away. In this way the comb gradually loses its charge.
Perhaps you could try it on different days. If the air is dry it should keep the charge for longer.