My daughter is prone to ear infections can wearing ear plugs in the shower help?

Depends. If the ear infection you are referring to is an external ear infection ("swimmer's ear"), then the plugs may help in the shower. However, if she is prone to middle ear infections, the plugs will serve no useful purpose in preventing middle ear infections.
Not really necessary. Your water should not harbor bacteria. You can use otc swimmers ear solution to dry it out, or mix your own with 1/3 alcohol, 1/3 vinegar and 1/3 water.