What I read is that a speaker produces sound by the movement of a coil attached to a cone which moves back and forth. So, If I try to move the coil by hand, would it produce sound? If not, why? or Why would it produce sound only when the current moves the coil?
Yes, the cone will produce sound waves if you move it to and fro by hand. The speaker produces sound by moving the cone, and it doesn't matter whether it's the moved by current in the coil or you moving it by hand. The sound waves will have whatever frequency you're moving the cone at, and I doubt you could manually move the cone at more than a few Hz, so the sound will have far too low a freequency for you to hear.
Also note that the efficiency of the sound generation falls off at low frequencies because it's hard for a small object to produce a sound wave with a large wavelength. Wiggling the cone by hand will produce a sound wave, but that wave will have a very low amplitude.