If I want Integrity and Authentication AND Non-Repudiation, can the Digital Signature be based on the MAC? Let's say I'm making a digital signature like this:

hash = hash(message)
digital signature = encrypt_with_private_key(hash)


where the private key is Alice's private key from a secure public-key cryptosystem.

And I'm making a MAC like this:

MAC = hash(message + some_secret_key)


And I then send both of these to my friend, encrypted using AES or some other cipher.

I feel I'm just creating work for him.

Could I simplify things like this?

MAC = hash(message + some_secret_key)
digital signature = encrypt_with_private_key(MAC)


Please tell me the pros and cons of this.
First of all, in a digital signature scheme that follows the hash-and-sign paradigm $Enc_{sk}(hash(m))$ the hash procedure is essential to "fit" the message $m$ in the public-key scheme domain: you cannot encrypt messages of any size with RSA for example. 

The second important point is that a digital signature scheme is a bit closed to public-key encryption: because we consider an untrusted channel to communicate the public-key; another similarity is that the signer wants the signature to be publicly verifiable. 

So, when we are using public-key cryptography, everything we do not want is to suppose a private/secure channel to exchange keys; you will need one if you want to use a cryptographic hash like a mac scheme. Furthermore, the signer has to exchange different secret keys with everyone who wants to verify the signature. Thus, creating distinct MAC tags: so, such a signature would no be publicly verifiable, but designated. 

Another important point is that care must be taken when a cryptographic hash is chosen: HMAC wasn't designed considering collision attacks M. Bellare, New Proofs for NMAC and HMAC: Security without Collision-Resistance. 
So, if you don't have a collision resistant mac scheme, what if a adversary can find another $tag'_i=mac(m_i)$, after consulting a polynomial number of $m_i, tag_i, Enc_{sk}(tag_i = mac(m_i))$, and so forging a signature? 

BTW, unforgeability is a cornerstone security property of digital signatures schemes.

Last but not least important. by using only a mac scheme and sharing key, we don't have non-repudiation or authentication. If the singer and verifier share a secret key, how can we prove which one created a tag? So mac isn't enough: you also need a public-key scheme.

Well...the pros... I can't see anyone. Sorry.