Are these problems NP-Complete?
Your first problem is $\mathsf{NP}$-Complete. Membership to $\mathsf{NP}$ is trivial. To see that it is $\mathsf{NP}$-hard, you can reduce it from Subset Sum. Let $\langle S_1, W_1 \rangle$ be an instance of Subset Sum. Define $S_2$ as the multiset obtained by multiplying each number in $S_1$ by $100$, and $W_2 = 100 W_1$. The instance for your problem is $\langle S_2, W_2 \rangle$.

If there is a set $C_1 \subseteq S_1$ such that $\sum_{x \in C_1} x  = W_1$, then let $W'$ be the sum of the elements in corresponding subset for $S_2$. We have $W' = \sum_{x \in C_1} 100x = 100 \cdot W_1 = W_2 \in \left[W_2-10, W_2+10\right]$.

If there is a set $C_2 \subseteq S_2$ such that the sum $W' = \sum_{x \in C_2} x$ of its elements satisfies $W_2-10 \le W' \le W_2+10$, then pick $y   = W' - W_2$, so that $W'  = W_2 + y$ and $y \in [-10, 10]$.
We must have $y=0$ since $W' = W_2 + y = y \pmod{100}$ and $W'$ must be a multiple of $100$. This shows that $W' = W_2$  and that the corresponding subset of $S_1$ has a sum of $\sum_{x \in C_2} \frac{x}{100} = \frac{W'}{100} = \frac{W_2}{100} = W_1$.

Your second problem is already known to be $\mathsf{NP}$-Complete even when when no vertex occurs in more than three triples. See problem SP1 in  "Garey &amp; Johnson, Computer and intractability: a guide to the theory of NP-completeness".