Should my dryer cord's neutral and ground be connected to the same terminal?
Sometimes you read something and can't help but shake your head. The answer is no, they wired the dryer wrong. The ground wire is supposed to wire to the ground lug of the dryer as shown in the manual on page 15.

The ground on the dryer needs to be wired correctly to provide safety for anyone who touches a metal surface if a short in the electrical system were to occur.