I read that the global dimension of $\mathbb Z/4\mathbb Z$ is not finite. I think that it's because that $4=2\cdot 2$ and $(2,2)\neq 1$, hence $\mathbb Z/2\mathbb Z\oplus \mathbb Z/2\mathbb Z$ is not $\mathbb Z/4\mathbb Z$.
Is it the reason for this ? If it's not too complicated, I would really like to see an explanation for why the global dimension is not finite.
It's not too complicated (I got it from Lam, Lectures on Modules and Rings, Lemma 5.16, who claims it is an idea of Kaplansky):
Consider the exact sequence $0\to\mathbb Z/2\mathbb Z\stackrel{2\cdot }{\to} \mathbb Z/4\mathbb Z\to \mathbb Z/2\mathbb Z\to 0$. Then $\mathbb Z/2\mathbb Z$ is not projective, since it is not a direct summand of $\mathbb Z/4\mathbb Z$ as you remarked. Therefore this yields that $\mathbb Z/2\mathbb Z$ has infinite projective dimension.