Null Characters and Splitting the String in the Pumping Lemma
Yes, it is possible for $x$ or $z$ to be 'null', if by 'null' you mean the empty string. The empty string, $\epsilon$, is the string of length zero. 

If the loop in the automaton (which is ultimately what the pumping lemma refers to) starts at the initial state and the initial state is also an accepting state, then strings of the form $y^i$ will be accepted, where $y$ is some string that will take you around the loop.

Whitespace plays no role in this theorem. Formal languages are defined over a given alphabet, in your case $\{0,1\}$. This is the only alphabet relevant for the question.  There is no preceding or trailing whitespace when talking about the pumping lemma. Spaces, tabs, return characters play absolutely no role.