whether the system is linear or not for the given problem
How can we calculate output at 0 input if the system depends on future or past or both?


Well, surely zero input just means;
$$
x(t) = 0~~~~\forall t
$$
and the $\forall t$ means for all time: positive and negative.

Substituting that into the equation:
$$
y(t) = x(t+1) + x(t-1) = 0 + 0 = 0
$$

So the system is homogeneous.

Well, as @Dilip points out, this isn't sufficient for homogeneity: we need the output to be $a y(t)$ for all inputs $a x(t)$. The case above just looks at $a=0$.

The next question is: does it satisfy additivity?

If 
$$
x_{\tt total}(t) = x_1(t) + x_2(t)
$$
then
$$
y_{\tt total}(t)  = x_{\tt total}(t+1) + x_{\tt total}(t-1)\\
= x_1(t+1) + x_2(t+1) + x_1(t-1) + x_2(t-1)\\
= y_1(t) + y_2(t)
$$
where 
$$
y_1(t) = x_1(t+1) + x_1(t-1)\\
y_2(t) = x_2(t+1) + x_2(t-1)
$$
so it satisfies additivity also.