How can you modify a SUBSET-SUM instance so evaluating a set outputs either 0 or 1? An SUBSET-SUM instance is a list of $n$ integers $\{ a_1, a_2,... a_n\}$. To evaluate a subset is to output the sum of a subset.
However, I want to know, is it possible to create a new instance $T$, the same size as the original set $S$, that any subset in $S$ that evaluates to $W$, the corresponding subset in $T$ (the numbers taken from the same positions) evaluates to $0$? All other subsets in $T$ should evaluate to $1$.
Bonus: Give a list of other NP-complete problems (other than 3SAT, where you evaluate a formula that either outputs $0$ or $1$ depending on the set of binary variables being passed into it), where evaluating an analogous instance outputs $0$ if it satisfies some objective related to the problem and $1$ otherwise.
However, I want to know, is it possible to create a new instance T, the same size as the original set S, that any subset in S that evaluates to W, the corresponding subset in T (the numbers taken from the same positions) evaluates to 0? All other subsets in T should evaluate to 1.

No, this is impossible. Consider $S=\{1,2,3\}$ and $W=3$. Our set $T$ should be $\{a,b,c\}$ such that $a+b=0, c=0$ and $a+b+c=1$.