Does getting my teeth bleached cause damage to the enamel on my teeth?

Nope. It should not cause damage at all to the enamel. It may make your teeth a bit more sensitive, but not damaging.
No. Bleaching is an acceptable process that causes teeth whitening. In some cases it can cause mild to moderate sensitivity for a brief period of time, which can be managed. It does not cause damage to tooth enamel.