Hint: First prove that $iLc$ satisfies the condition in 3, where $i:D\to C$ is the inclusion. Then use apply 2 to conclude.
Regarding "localization" the point is that $L$ is characterized by the arrows it inverts, so the language is being imported from commutative algebra. Specifically, the motivating situation is that of $R[S^{-1}]$-modules, where $S$ is a multiplicative set in a commutative ring $R$. Every $R[S^{-1}]$ is functorially an $R$-module by restricting the scalars, and this functor is fully faithful, with the left adjoint $R[S^{-1}]\otimes_R (-)$. So this is a reflective subcategory. Furthermore an $R$-module admits an $R[S^{-1}]$-action if and only if it is local for those $R$-module maps inverted by tensoring with $R[S^{-1}]$, as in Riehl's point 3. The classical algebraic framework would reduce those maps to the maps $s:R\to R$ determined by elements of $S$, while the general categorical framework instead asks for locality with respect to all maps $\eta_A:A\to A[S^{-1}]$, for $A$ an $R$-module. We can bridge these frameworks by observing that locality with respect to each $s$ is equivalent to locality with respect to the single unit map $R\to R[S^{-1}]$, which implies locality with respect to every $\eta_A$ by considering the action of $L$ on a presentation of $A$.
The subcategory of (path-)connected spaces is not coreflective. To show that the subcategory of locally (path-)connected spaces is coreflective, it suffices to check that the discrete topology is locally (path-)connected, and that the intersection of locally (path-)connected topologies is still locally (path-)connected.
A full subcategory $\mathcal C$ of topological spaces is coreflective if for each topological space $X$ has a coreflector, that is, a continuous map $CX\to X$ from $CX$ in $\mathcal C$ such that every continuous map $Y\to X$ for $Y$ in $\mathcal C$ factors as $Y\to CX\to X$ for a unique continuous map $Y\to CX$.
Note that if $X$ is in $\mathcal C$ then for $CX=X$ with the identity morphism is such a coreflector.
Note also that given two coreflectors $CX\to X$ and $CY\to Y$ and a continous map $X\to Y$, the composite $CX\to X\to Y$ must factor through $CY\to Y$ via a unique continuous map $CX\to CY$. It turns out that a choice of coreflectors for each topological space, together with the corresponding association of morphisms, is exactly the data of a right adjoint to the inclusion functor.
If $\mathcal C$ is the category of (path-)connected spaces and $CX\to X$ is a coreflector, then on the one hand the image of $CX\to X$ is contained in one (path-)component of $X$, and on the other hand the inclusion $C\hookrightarrow X$ of every (path)-component $C$ of $X$ into $X$ ha sto factor through $CX\to X$, whence must lie in the (path-)component of $X$ containing the image of $CX$. Thus, $X$ has only one (path-)component, i.e. is (path)-connected.
Thus the subcategories of (path-)connected space are not coreflective.
In the case of locally (path-)connected topological spaces, this amounts to verifying that for every topological space $X$ has a coarsest locally (path)-
connected topology finer than it.
Given all continuous functions $Y\to X$ for $Y$ in $\mathcal C$, there is a coarsest topology on $X$ for which they are continuous. This topology is at least as fine as that of $X$, so we have a continuous map $CX\to X$ (the identity set-function changing to a possibly coarster topology) through which every continuous function $Y\to X$ for $Y$ in $\mathcal C$ factors uniquely.
It follows that a sufficienct (but not necessary) condition for the subcategory to be coreflective is that the above-defined $CX$ be in $\mathcal C$ for each topological space $X$. In particular, it suffices for the discrete topologies to be in $\mathcal C$ and for the intersection of topologies in $\mathcal C$ on the same set to also be a topology in $\mathcal C$.
Best Answer
Your intuitive description of reflective subcategories doesn't distinguish left adjoints from right adjoints so you could apply it equally well to either. Anyway, I think the best way to get a handle on these sorts of things is through a bunch of examples.
Examples of reflective subcategories:
Examples of coreflective subcategories:
These algebraic examples work loosely as follows. A reflective subcategory consists of "nice" objects such that every object has a maximal "nice quotient" (with reflector given by this quotient), and a coreflective subcategory consists of "nice" objects such that every object has a maximal "nice subobject" (with coreflector given by this subobject). Probably there are examples a bit stranger than this (for example one could take the opposite category of metric spaces or topological spaces) but I think this is a decent place to start.