Given that $a,b$ and $c$ are integers.
and $\log \mid a \mid + \log \mid b \mid + \log \mid c \mid =0$
we know that $\log \mid a \mid + \log \mid b \mid + \log \mid c \mid =\log(\mid a \mid\cdot\mid b\mid\cdot\mid c\mid)$
$\implies\log(\mid a \mid\cdot\mid b\mid\cdot\mid c\mid)= 0$
$\implies \mid a \mid\cdot\mid b\mid\cdot\mid c\mid={10}^{0}$
$\implies \mid a \mid\cdot\mid b\mid\cdot\mid c\mid=1$
For minimum value of $a+b+c$ we can put $a=b=c=-1$ and get
$(a+b+c)_{\min}=-1-1-1=-3$
For maximum value of $a+b+c$ we can put $a=b=c=1$ and get
$(a+b+c)_{\max}=1+1+1=3$
So, (A) is the correct answer.