In our current Measure Theory Class, we bought up the notion for a function $f:\mathbb R \to \mathbb R$ that is continuous to have a compact support, is equivalent to the fact that $\overline{\{x \in \mathbb R: f(x)\neq 0\}}$ needs to be compact.
Note $C_{c}(\mathbb R)=\{ f: \mathbb R \to \mathbb R:f$ is continuous and $supp f$ compact $\}$
I assume the most important aspect to take out of the notion of compact support (at least for my class in Measure Theory) is the fact that: $\forall p \in [1,\infty]:$
$C_{c}(\mathbb R)\subseteq L^{p}$
So if I need to prove a function is $p-$integrable then I can simply show that it has compact support $supp f$.
Is this the most applicable case to use the notion of a compact support in Measure Theory?
Other Question:
Considering the factor $\partial \{x \in \mathbb R: f(x)\neq 0\}$, does this mean that the set $\overline{\{x \in \mathbb R: f(x)\neq 0\}}$ can contain points $y \in \mathbb R$ such $f(y)=0$ but these sets of points need to be countable? Or can they be uncountably infinite?
In measure theory in isolation the concern is really more about finite measure support (where support has no closure operation in that setting, in part because there is no guarantee that a topology has been defined on the measure space). The point there is that a bounded measurable function with finite measure support is in all of the $L^p$ spaces. Compact support is important for other reasons but it starts to involve topics that are not strictly measure-theoretic in nature (e.g. distribution theory).
Your other question (how "big" can the boundary of the support of a continuous function be?) should probably be asked separately.