“Can local anesthesia cause nerve damage?”

Yes it could happens. Direct stimulation of a peripheral nerve during local anesthesia injection can cause direct nerve perforation. Nerve injuries affected by the size and the type off the needle.

Theoretically it could. When injected in high concentration directly into the nerve. However with the current technology utilizing ultrasound its very unlikely, since you can visualize an injection location.

Local anesthetic, in and of self as an entity, probably not. The placement of it (with a needle) possibly. The question specifically asked about 'nerve' damage, so no, but the local anesthetic itself, possibly--injected into a blood (heart issues) vessel, too much local anesthetic--problems, allergic reaction to the particular compound--problems. Each with those things being said, anesthesia-- from a purely statistical perspective, is a much safer experience than turning the keys in the ignition of your car and driving.

Normally, no.