My second question was not intended to call into question the validity of the law of conservation of momentum, but rather the validity of the system it was being applied on. I thought that, since some of the particles were interacting with the air, the surrounding environment would have to be considered part of the system as well as the pendulum and bullet to account for the momentum lost to the air (I understand now that the system is just fine defined the way it is).
I have read your replies and thought about this some more, and I think that, when the bullet and pendulum collide, this "lost" momentum I was referring to is the result of some of the kinetic energy turning into increased molecular motion. But, as Russ said much earlier, the motion of the molecules would be totally random, and the momentum from the vibrations would be zero as a whole. So even though the object would indeed be interacting with the air molecules, causing them to vibrate more vigorously and thus increasing their average kinetic energy, it would be doing so symmetrically. There would be no momentum lost to the environment; all those little mv's the molecules had would add up to zero, but kinetic energy lost would not be zero, as many of you have said, since it is a scalar.
When I said radiating heat, I should not have. The sense I meant it in is not in line with the definition used in physics. In all these discussions, when I mentioned 'radiated', I meant the vibration of the object's molecules would be interacting with the surrounding air molecules, increasing the air molecules movement, while decreasing the movement of the object's molecules. So, in a sense, they are "giving off" motion, but because this motion is not in anyone direction, but spreads out evenly in all directions, increasing the average kinetic energy of the surrounding air and thus its temperature, the air's momentum has not actually increased. It has not increased because all the random movements have to balance out to zero. So all the momentum is still within the bullet pendulum system. This same idea would apply to the refrigerator.
Though this lost kinetic energy could not by itself cause motion in anyone direction, it could affect other objects. For instance, if the experiment were done under water, the heat generated might cause water to undergo a phase change, if the kinetic energy produced were large enough, but it could not cause water to move in any specific direction, it would not increase the momentum the water had before the collision.
Everyone else who replied to this topic, thank you for your time and consideration, all of the replies here helped me to think about the situation through other eyes. Kinetic energy and momentum are completely different things. I had learned that but not really understood it until now, with the help of many of you whom have taken the time to point out the errors in my argument and to offer your own arguments, in some cases. I feel much more confident in my understanding of momentum and kinetic energy, as they apply to these two situations and as a whole.
To answer my own questions with a better understanding:
The refrigerator would not move, except maybe imperceptibly due to photon radiation, if some of those photons were somehow blocked so that there were a net direction in which this radiation pressure occurred. The momentum of the coil would remain constant, even while it gave off kinetic energy to the air.
The bullet-pendulum system's velocity will not be affected due to loss of what is really just kinetic energy, but what I continued, in error, to call momentum.
-Tony