Proving security of $\Pi' = E_k(E_k(m))$ knowing $\Pi = E_k(m)$ is secure
As Mikero noticed, the problem in your first proof is the fact that you can't suppose that $\mathcal{A}$ knows the secret $k$, and thus it can not encrypt the challenge.
And if you don't encrypt the challenge, then the input is not what is expected by $A'$. $A'$ is waiting $Enc^2_k(m)$ or $Enc^2_k(m')$ as a challenge not $Enc_k(m)$ or $Enc_k(m')$.
When you are doing a game-based proof, an important notion is the notion of
indistinguishability. For example If you are using an algorithm $\mathcal{A}$ as an oracle. And if you want to use some properties about the output of $\mathcal{A}$, it is important to check that the input of $\mathcal{A}$ follow the distribution which is mentioned in the property.
In your example, you are giving to the adversary $Enc(m)$ and not $Enc(Enc(m))$, it's completely different in the general case.
Let consider the xor encryption $Enc_k(m)= k\oplus m$.
It's really easy to build an attacker (let say a powerfull attacker who decrypts the ciphertext without the secret key) against $Enc^2$, it's just the identity function (Because $Dec_k^2=Enc^2_k$ is the identity function for all $k$.)
Then even $Enc$ is semantically secure, then $Enc^2$ could not to be. So a) is false.
But your proof about (b) seems correct to me.