Clarification on definition of a Sheaf

1. A (pre)sheaf on a category $\mathcal{C}$ is a functor from $\mathcal{C}^{\rm op} \to \mathsf{Set}$ or sets with extra structure: Abelian groups, rings, modules, etc. $\mathcal{C}$ is often the category of open sets on a topological space. In particular, $\mathscr{F}(U)$ is always a set, by definition.

2. In the equation $\operatorname{res}_{V \cap W }(s_i) = \operatorname{res}_{V \cap W }(s_j)$ we assume that $s_i \in \mathscr{F}(V)$ and $s_j \in \mathscr{F}(W)$. Or to use the notation on Wikipedia: $s_i \in \mathscr{F}(U_i)$ and $s_j \in \mathscr{F}(U_j)$ with $$ \operatorname{res}_{U_i \cap U_j}(s_i) = \operatorname{res}_{U_i \cap U_j}(s_j) $$ For a skyscraper sheaf (of sets, let's say) $\mathscr{F}(U) = \{0\}$ (the terminal object of $\mathsf{Set}$ up to isomorphism) for all $U$ not containing $p$ and hence $s_i = 0$ for all $U_i$ not containing $p$. These $s_i$ still exist.

Maybe it would be best for you to read some examples of sheaves and think through the glueing axiom. For example, the sheaf of continuous functions on a topological space or sheaves of smooth/continuously differentiable functions on a manifold.

I assume you are considering $U$ to be an open set in some topological space, so your category is $\mathrm{Top}(X)$, and your presheaf is a functor $\mathscr{F}\colon\mathrm{Top}(X)^{\text{op}}\to \mathrm{Set}$. To be able to discuss "glueing", your source category must have a notion of "cover" of an object, that is a Grothendieck topology (such a category is called a site). In such context, the sheaf condition is expressed by an equalizer diagram for any cover. While this approach requires some more machinery, it allows to express the glueing axiom without having to evaluate the restriction maps at sections.