Minimum pumping length of (01)*
The minimum pumping length $\ell$ of $(01)^*$ is $2$.
First of all notice that there are no words of length $1$ in the language.
Then, if you look at a formal definition of the pumping lemma for regular languages (what you wrote is still informal) you'll find that the length $|xy|$ of the prefix $xy$ of a word in the language that can be pumped must be $\le \ell$.
This allows you to prove that the minimum pumping length $\ell$ must be $2$ by showing:

That there is a DFA for $(01)^*$ that uses at most $2$ states, implying $\ell \le 2$. This DFA is trivial to obtain.
That $\ell=1$ is not a valid pumping length. This is true since $|y|$ must be positive and, together with $|xy| \le \ell$, this implies $|y|=1$. In other words $x = \epsilon$ and $y=0$. It is easy to obtain a word not in the language by repeating $y$ either $0$ or at least $2$ times.