You associate to each topological space $X = (X,T)$ a certain topology $T_\ell$ on the set $X$ and call the space $X_\ell = (X,T_\ell)$ the locally connected modification of $X$. In your class it was proved that $X_\ell$ is locally connected.
Before we come to your four questions, let us try to understand the locally connected modification from a more conceptual point of view.
By an lcd-map we mean any continuous map $f : Y \to X$ with a locally connected domain $Y$. By an elementary lcd-map we mean any lcd-map of the form $\iota = id_X : (X,T') \to X$ with a topology $T'$ on the set $X$. The elementary lcd-maps can be identified with the topologies $T'$ on the set $X$ such that $T'$ is finer than $T$ and $(X,T')$ is locally connected. Let us call such a topology an lc-refinement of $T$. Note that the discrete topology is always an lc-refinement.
A locally connected modification of a space $X$ is now be defined as an lcd-map $\lambda : X_\ell \to X$ having the following universal property:
For each lcd-map $f : Y \to X$ there exists a unique continuous $f_\ell : Y \to X_\ell$ such that $\lambda \circ f_\ell = f$.
Note that $f_\ell$ is an lcd-map with a locally connected codomain $X_\ell$. We often write locally connected modifications of $X$ in form of pairs $(X_\ell,\lambda)$.
If $X$ is locally connected, then trivially $id_X : X \to X$ is a locally connected modification of $X$. But what if $X$ is not locally connected? In that case it is not yet clear whether $X$ has a locally connected modification and what the "new" definition has to do with your definition of $X_\ell$ via the explicitly given topology $T_\ell$.
To study this relationship, let us call a topology $T_\ell$ lc-universal if it is an lc-refinement of $T$ such that the elementary lcd-map map $\iota_\ell = id_X : (X,T_\ell) \to X$ is a locally connected modification of $X$.
A natural question is whether the existence of a locally connected modification implies the existence of an lc-universal topology on $X$.
- Locally connected modifications of $X$, if they exist, are unique up to canonical homeomorphism.
Let $\lambda : X_\ell \to X$ and $\lambda' : X'_\ell \to X$ be locally connected modifications. The universal property gives a unique $\lambda'_\ell : X'_\ell \to X_\ell$ such that $\lambda \circ \lambda'_\ell = \lambda'$ and a unique $\lambda_\ell : X_\ell \to X'_\ell$ such that $\lambda' \circ \lambda_\ell = \lambda$. We conclude that $\lambda \circ (\lambda'_\ell \circ \lambda_\ell) = \lambda$. But trivially $\lambda \circ id_{X_\ell} = \lambda$, thus by the universal property $\lambda'_\ell \circ \lambda_\ell = id_{X_\ell}$. Similarly $\lambda_\ell \circ \lambda'_\ell = id_{X'_\ell}$. This shows that $\lambda_\ell$ and $\lambda'_\ell$ are "canonical" homeomorphisms which are inverse to each other. $\square$
- If $(X_\ell,\lambda)$ is a locally connected modification and $h : X'_\ell \to X_\ell$ is a homeomorphism, then $(X'_\ell,\lambda \circ h)$ is a locally connected modification.
This is obvious.
- There exists at most one lc-universal topology on $X$.
Consider lc-universal topologies $T_\ell, T'_\ell$ on $X$. They give elementary locally connected modifications $\iota = id_X : (X,T_\ell) \to X$ and $\iota' = id_X : (X,T'_\ell) \to X$. The proof of 1. produces a homeomorphism $\iota_\ell : (X,T_\ell) \to (X,T'_\ell)$ such that $\iota' \circ \iota_\ell = \iota$. Since $\iota, \iota'$ are the identities on the level of sets, also $\iota_\ell$ is the identity on the level of sets and hence $T_\ell = T'_\ell$. $\square$
To prove that each $X$ has a locally connected modification, we need a construction. The universal property involves all locally connected spaces $Y$ and all maps $Y \to X$. From the conceptual point of view we need all of them; they enter in the characteristic universal property.
As you say, this is a "huge class". Indeed, the lcd-maps $f : Y \to X$ do not form a set, but a proper class.
However, it is definitely not expedient to restrict to a single locally connected space $Y$. What would be a "good choice" here? For example, taking a discrete space $Y$ does not make much sense. In the universal property we cannot a priori exclude spaces $Y$.
- If $\lambda : X_\ell \to X$ is a locally connected modification of $X$, then $\lambda$ must be bijective.
Let $X_d$ denote the set $X$ with the discrete topology. Since $X_d$ is locally connected, there exists $i': X_d \to X_\ell$ such that $\lambda \circ i' = id_X$, and this is possible only when $\lambda$ is surjective.
Now let $\lambda(\xi_1) = \lambda(\xi_2) = x \in X$. The space $Y = \{0\}$ is locally connected and $f : Y \to X, f(0) = x$, is continuous. Define $f_j : Y \to X_\ell, f_j(0) = \xi_j$. Then $\lambda \circ f_j = f$, thus $f_1 = f_2$ which implies $\xi_1 = \xi_2$. $\square$
- $X$ has a locally connected modification if and only if it there exists an lc-universal topology on $X$.
Let $(X_\ell,\lambda)$ be a locally connected modification. Since $\lambda$ is a bijection, there exists a unique topology $T_\ell$ on $X$ such that $\lambda : X_\ell \to (X,T_\ell)$ is a homeomorphism. Then $\iota = id_X : (X,T_\ell) \xrightarrow{\lambda^{-1}} X_\ell \xrightarrow{\lambda} X$ is a locally connected modification. $\square$
- $T_\ell$ is an lc-universal topology if and only if all lcd-maps $f : Y \to X$ are continuous as maps $Y \to (X,T_\ell)$.
In fact, the universal property reduces to the above condition. On the level of sets $f : Y \to (X,T_\ell)$ is the only function such that $\iota_\ell \circ f = f$, thus it must be continuous. $\square$
The above considerations show that the only candidate for an lc-universal topology is the final topology $T_\ell$ with respect to the sink of all lcd-maps. Explicitly
$$U \in T_\ell \Longleftrightarrow \text{ For all lcd-maps } f: Y \to X, f^{-1}(U) \text{ is open in } Y . \tag{1}$$
In fact, we know that $T_\ell$ is an lc-universal topology if and only if all lcd-maps $f : Y \to X$ are continuous as maps into $(X,T_\ell)$. The latter means that for all lcd-maps $f : Y \to X$ all preimages $f^{-1}(U)$ with $U \in T_\ell$ are open in $Y$. $\square$
The above topology is that defined in your question, but there seems to be a problem here: The lcd-maps $f : Y \to X$ with locally connected $Y$ do not form a set. We therefore do not get a sink in the usual interpretation (this requires that the maps into $X$ are indexed by a set).
But actually it is no problem to generalize the concept of a sink to families of functions indexed by classes. See for example Chapter 0 Section 2.2 "Classes" in
Adámek, J., Herrlich, H., & Strecker, G. (1990). Abstract and concrete categories. Wiley-Interscience.
Anyway, defining a topology $T_\ell$ via $(1)$ is not problematic (one does not need to know what a sink and a final topology is).
However, we can strongly downsize the "huge class" of all lcd-maps $Y \to X$ to something more manageable. Tyrone has given hints in a comment.
- $\lambda : X_\ell \to X$ is a locally connected modification of $X$ if and only if the universal property is satisfied for all surjective lcd-maps $f : Y \to X$.
Consider an arbitrary lcd-map $f : Y \to X$. The disjoint union $Y + X_d$ is locally connected. Define $F : Y + X_d \to X$ by $F \mid _Y = f$ and $F \mid_{X_d} = id_X$. This is a continuous surjection so that we find a unique $F_\ell : Y + X_d \to X_\ell$ such that $\lambda \circ F_\ell = F$. Let $f_\ell = F_\ell \mid_Y$. Then $\lambda \circ f_\ell = f$. Given any $f' : Y \to X_\ell$ such that $\lambda \circ f' = f$, we define $F' : Y + X_d \to X_\ell$ by $F' \mid_Y = f'$ and $F' \mid_{X_d} = F_\ell \mid_{X_d}$. Then $\lambda \circ F' = \lambda \circ F_\ell$, hence $F' = F_\ell$ and $f' = f_\ell$. $\square$
- $\lambda : X_\ell \to X$ is a locally connected modification of $X$ if and only if $\lambda$ is bijective and the universal property is satisfied for all bijective lcd-maps $f : Y \to X$. It even suffices to require that the universal property is satisfied for all elementary lcd-maps $\iota = id_X :(X,T') \to X$.
Consider an arbitrary continuous surjection $f : Y \to X$. Let $X_f$ be the set $X$ endowed with the quotient topology induced by $f$. The space $X_f$ is locally connected; see A quotient of a locally connected space is locally connected. The identity $\iota = id_X : X_f \to X$ is continuous (which means that the topology of $X_f$ is finer than $T$ and hence an lc-refinement of $T$). Indeed, if $U \subset X$ is open, then $f^{-1}(U)$ is open in $Y$, hence $U = id_{X}^{-1}(U)$ is open in $X_f$. We have $f = \iota \circ \varphi$ with $\varphi = f : Y \to X_f$. We know that there exists a unique $\iota_\ell : X_f \to X_\ell$ such that $\lambda \circ \iota_\ell = \iota$. Hence $f_\ell = \iota_\ell \circ \varphi$ satisfies $\lambda \circ f_\ell = f$. Given any $f' : Y \to X_\ell$ such that $\lambda \circ f' = f$, we get $\lambda \circ f' = \lambda \circ f_\ell$. Since $\lambda$ is a bijection, $f' = f$. $\square$
This shows that definition $(1)$ is equivalent to
$$U \in T_\ell \Longleftrightarrow \text{ For all elementary lcd-maps } \iota = id_X : (X,T') \to X, U = \iota^{-1}(U) \in T' . $$
In other words, $T_\ell$ is the intersection of all lc-refinements $T'$ of $T$.
As you know, $(X,T_\ell)$ is locally connected, thus $T_\ell$ is the unique lc-universal topology on $X$.
Let us close with a bit category theory. By $\mathbf{Top}$ we denote the category of topological spaces and continuous maps , and by $\mathbf{LC}$ the full subcategory of locally connected spaces.
What we have done so far means that the inclusion functor $\mathbf{LC} \to \mathbf{Top}$ has a right adjoint. This can expressed in the form "$\mathbf{LC}$ is a coreflective subcategory of $\mathbf{Top}$". See Theorem 4.1 here.
The existence of a right adjoint is by no means trivial. For example, the inclusion of the category $\mathbf{Haus}$ of Hausdorff spaces into $\mathbf{Top}$ does not have a right adjoint. See inclusion functor Haus $\rightarrow$ Top does not have a right adjoint.
