Internal friction in the metal of the bell eventually will bring the ringing vibrations to an end.
The bell vibrates when it rings, making its molecules more energetic and creating heat. Bonding between the molecules of the bell resist the vibrations, and eventually the strength of the molecular bonds will create enough friction to bring the vibrations to an end.
To address your 2nd question, see this account of a project to dampen machine vibrations by applying a magnetic field to a tool holder: http://dynamicslab.mpe.nus.edu.sg/dynamics/Project0506/thesis0506/Vibration%20damping%20using%20magnetic%20field%20boring%20process.pdf It was found that a magnetic field will dampen vibrations of steel, which is both electorally conductive and magnetic, but is less effective in non-magnetic metals such as aluminum, and is absent in non-metallic substances. Interestingly, though it is theoretically possible for a magnetic field to dampen vibrations, most of the noise dampening effect arose fromachieved in the project was due to the mass of thean electromagnet attached to the apparatus. holder!