Is authenticated encryption unnecessary if the message is digitally signed? Normally, it is necessary to use authenticated encryption if the message may be tampered with.  Can the authentecation be omitted if the message has been otherwise authenticated?
Yes, message authentication can be omitted when using an appropriate digital signature on the ciphertext.

A digital signature hashes the message, then encrypts the hash with a nonce and asymmetric key, which makes the security similar to HMAC.

The reason authenticated encryption is preferred is that it is generally less computationally expensive than a digital signature, and easier to build a MAC construct that is side channel resistant.