Question 1:
For a discrete group $G$, does $\chi(G)$ denote the Euler characteristic of $G$, or is this notation commonly used for something else?
For the next three questions, assume $\chi$ does denote Euler characteristic. (I am asking to check on some references.)
Question 2:
Is it the case that if $G$ is a finite group, then $\chi(G) = 1/|G|$?
An affirmative answer to question 2 is stated at the bottom of the generalizations section here.
Question 3:
Suppose $G$ and $H$ are groups such that $\chi(G)$ and $\chi(H)$ exist, does $\chi(G * H) = \chi(G) + \chi(H) - 1$?
If it is the case that $\chi(G)$ exists implies that $G$ has a finite classifying space, then an affirmative answer to question 3 is given on page 46 here.
Question 4:
Is $\chi(\mathbb{Z}) = 0$?
This is stated here, and I think can be seen since the circle $S^1$ is the classifying space of the infinite cyclic group, and the circle consists of one 0-cell and one 1-cell, and 0 = 1 - 1.
Of course, if the answers to all the above questions are yes, then we would have that $$ \begin{aligned} \chi(\mathbb{Z} * \mathbb{Z}/m\mathbb{Z}) &= \chi(\mathbb{Z}) + \chi( \mathbb{Z}/m\mathbb{Z}) - 1\\ &= 0 + \frac{1}{m} - 1\\ &= \frac{1}{m} - 1. \end{aligned} $$