Speaking for myself, I'm not very fond of the "structural" view of pure mathematics which your classification 1 through 4 suggests. It is not just that
the different structures can be combined, but that the same idea can be encoded in apparently different structures.
E.g. Klein's Erlangen program puts forward the point of view that geometries should be described in terms of the symmetry groups that act on them. From a modern point of view, this leads (among other things) to the notion of symmetric spaces, which are fundamental objects in many areas of modern mathematics, from topology to mathematical physics to Lie theory.
E.g. In number theory, the property of congruence of numbers (in the sense of
modular arithmetic) can be encoded in terms of considering the different possible metric completions of the field $\mathbb Q$ (this is Ostrowski's theorem).
Regarding your question about other structures, you probably know that dynamical systems is a thriving area of pure mathematics. Many different structures can appear there: discrete or Lie group actions on manifolds or measure spaces, cocycles, objects from descriptive set theory, etc. Like most active areas of mathematics, thinking in terms of structures probably isn't the best way to understand the goals of and developments in the subject.
I don't mean to downplay the importance of structure; I just don't think that it provides the preeminent lens through which to view modern pure mathematics.
The heart of it is that, while your intuition is that "continuity" is about distances - you first learn continuity in terms of $\epsilon-\delta$ proofs - it turns out that continuity is really only about open sets. Whether a map $f:X\to Y$ between metric spaces is continuous is entirely determined by the open sets of $X$ and $Y$. If you have two different metrics for $X$ which determine the same open sets, the set of continuous functions on $X$ are the same.
Since topology is the study of continuity, it makes sense to only care about open sets, then, not metrics.
The original definitions for "topology" had more rules about the open sets that made topologies "more like" metric spaces. But as mathematicians started playing with these things, they found that it made sense to ask questions about continuity in the cases where these rules were broken, too. So the definition was broadened to give the widest meaning.
A very basic example might be left-continuity. A function $f:\mathbb R\to Y$ with $Y$ a metric space is called left-continuous if $\lim_{x\to a-} f(x)=f(a)$ for all $a\in\mathbb R$.
It turns out, there is a topology on the real line, call it $\tau_{L}$, under which a function $f$ is left-continuous if and only if it is (topologically) continuous as a function from $(\mathbb R,\tau_L)\to Y$. (Note: $\tau_{L}$ has a basis the intervals of the form $(x,y]$.) Now, $\tau_{L}$ is a point-set topology, but it is not one that comes from a metric. So the notion of "left-continuity" is an example of an idea that fits the point-set definition of continuity, but does not fit the $\epsilon-\delta$ notion of continuity - you essentially need to redefine it.
The fact that left-continuity and right-continuity together is the same as "normal" continuity is a statement about the three different topologies. Somehow, the usual real line topology is a combination of these two other topologies.
It turns out there is something really deep going on here. Somehow, the open neighborhoods of a point in a topology contains a lot of information - what we call "local information" - about behavior of "nice" functions at that point.
Best Answer
The central idea (as is true of much of Mathematics) is to generalize some very familiar concepts to more abstract notions. In particular, there are some things that the notions of length, area, volume, etc. have in common. For one example, if you have nothing, then what you have has no length, area, or volume. For another example, if you have two (or more, up to countably-infinitely-many) distinct objects with clearly-defined length/area/volume, then what you have should also have a clearly-defined length/area/volume, found by simply adding the lengths/areas/volumes of the individual objects.
The purpose of Measure Theory is to capture these common characteristics, and allow them to be applied to versions of "measure" that are unspecified or less intuitive/visualizable. In particular, it lets us (attempt to) answer certain questions, such as: "Can we measure every subset of the real line in a way that corresponds with length for intervals and acts like measurements 'should'?"
It turns out that the answer is: "Well...maybe...if we take certain things for granted." So, what seems like a simple question turns out to be very deep, indeed! In fact, in order for certain principles of measure theory to make sense, one must take certain things for granted. For example, if one doesn't assume that a countable union of disjoint finite sets is countable, then it is completely consistent with Measure Theory that the real number line is of length $0$! If you're curious about the details of the crazy-sounding claims, let me know, and I will explain and/or provide links to clear it up.