$\left(1\right)\because$当$x=-2$时，分式$\frac{x-b}{x+a}$无意义，当$x=4$时，此分式的值为零，
$\therefore \left\{\begin{array}{l}{-2+a=0}\\{4-b=0}\\{4+a≠0}\end{array}\right.$，
解得$\left\{\begin{array}{l}{a=2}\\{b=4}\end{array}\right.$，
$\therefore a+b$的值为$6$；
$(2)\because$分式$\frac{6}{x-2}$的值是整数，
$\therefore x-2=-6$，$-3$，$-2$，$-1$，$1$，$2$，$3$，$6$，
$\therefore x=-4$，$-1$，$0$，$1$，$3$，$4$，$5$，$8$.

$\left(1\right)\because$当$x=-2$时，分式$\frac{x-b}{x+a}$无意义，当$x=4$时，此分式的值为零，
$\therefore \left\{\begin{array}{l}{-2+a=0}\\{4-b=0}\\{4+a≠0}\end{array}\right.$，
解得$\left\{\begin{array}{l}{a=2}\\{b=4}\end{array}\right.$，
$\therefore a+b$的值为$6$；
$(2)\because$分式$\frac{6}{x-2}$的值是整数，
$\therefore x-2=-6$，$-3$，$-2$，$-1$，$1$，$2$，$3$，$6$，
$\therefore x=-4$，$-1$，$0$，$1$，$3$，$4$，$5$，$8$.