Is this problem same as discrete logarithm? Given $g,h\in\mathbb Z_p$ where $g$ generates $\mathbb Z_p^\star$ Discrete logarithm problem is to find $z$ such that $g^z\equiv h\bmod p$ holds.

Take the problem given $g,g',h$ where $g^z\equiv h\bmod p$ holds at some $z$ find $h'$ such that $g'^{z}\equiv h'\bmod p$ holds.

If we know $y$ such that $g^y\equiv g'\bmod p$ then $g'^{z}\equiv h'\bmod p$ is known or if we know $z$ then $g'^{z}\equiv h'\bmod p$ is known.


If we know $g'^{z}\equiv h'\bmod p$ and $g^z\equiv h\bmod p$ then is it easier to find $z$?
What is the best way to find $g'^{z}\equiv h'\bmod p$ if we only know $g,g',h$ with $g^z\equiv h\bmod p$?
What is the best way to find $f$ such that $f^{z}\equiv 1\bmod p$ if we only know $g,h$ with $g^z\equiv h\bmod p$?
This is exactly the Computational Diffie-Hellman (CDH) problem. Rename $h=g^z$ and $g'=g^y$ in what you wrote, then the problem becomes:


  Given $g$, $g^z$, and $g^y$, compute $g^{yz}$.


See this discussion to address your question about the relationship between CDH and discrete log.

edit: 


  Q1: If we know $(g′)^z$ and $g^z$ then is it easier to find $z$?


No, it is just as hard as discrete log. If you are given just $g$ and $g^z$ then you can generate other pairs with the same discrete log relationship yourself, via $g' = g^r$ and $(g')^z = (g^z)^r$. So having extra such pairs doesn't add any new information.