Usage of pairings in proxy re-encryption algorithm
Is $Z^{ak}$ the same as $e(g^a,g^k)=e(g,g)^{ak}$?


That's correct: by the bilinearity property of the pairing $Z^{ak}=e(g,g)^{ak}=e(g^a,g^k)$.


  And is $mZ^k$ the same as $e(g^k,g^k)=e(g,g)^{k^2}$?


No it is not. $mZ^k$ is the group operation of $\mathbb{G}_2$ applied to $m$ and $Z^k$.


  I would like to understand the intermediate steps like I have for an ElGamal.


The first-level decryption algorithm, given a ciphertext $c_a=(\alpha,\beta)$, as per their specification involves computing $\beta/\alpha^{1/a}$. If the ciphertext is properly formed then $\alpha=Z^{ak}$ and $\beta=mZ^k$ and therefore 
  $$\frac{\beta}{\alpha^{1/a}}=\frac{mZ^k}{(Z^{ak})^{1/a}}=\frac{mZ^k}{Z^k}=m.$$