Pumping lemma for 0^n and n>0
The pumping lemma says that you can split the word into $xyz$. It doesn't say that every such split works. For instance, if you take $|x|=1$ in your example, the pumping works up and down. This would be a "correct split" that makes the pumping lemma hold. 

As a reference, recall the definition of the pumping language for regular languages (from Wikipedia): 


  Let $L$ be a regular language. Then there exists an integer $p\ge  1$ (depending only on $L$) such that every string $w$ in $L$ of length at least $p$ ($p$ is called the "pumping length") can be written as $w = xyz$ (i.e., $w$ can be divided into three substrings), satisfying the following conditions: 
  
  
  $|y| \ge 1$  
  $|xy| \le p$ and  
  for all $i \ge 0$, $xy^iz \in L$.
  $y$ is the substring that can be pumped (removed or repeated any number of times, and the resulting string is always in $L$).