Can I sign 'arm' on either side of my body?
It is standard practice to use your dominant hand to trace your non-dominant arm. For example, if you are right-handed, you should use your right index finger to indicate your left arm.
Does the direction of the tracing motion matter?
Generally, tracing from the upper arm down toward the wrist is the most natural and common way to sign it, but simply indicating the length of the arm in either direction is easily understood.
Is this sign used for animal limbs as well?
No, this specific sign refers to a human arm. When talking about animals, you would typically use classifiers or specific signs for paws, legs, or wings depending on the creature.