I work with infinite dimensional manifolds so am extremely distrustful of anything that requires some sort of compactness condition. Most of the time, it's just too restrictive.
Consider a really nice simple space: the coproduct of a countably infinite number of lines, $\sum_{\mathbb{N}} \mathbb{R}$ (coproduct taken in the category of locally convex topological vector spaces). This has the property that any compact subset is contained in a finite subspace. However, any neighbourhood of the origin has to be absorbing (the union of the scalar multiples of it is the whole space) so there aren't any non-zero continuous functions with compact support. That defenestrates option 4.
Particularly simple functions on an infinite dimensional vector space are the cylinder functions. These are important in measure theory on such spaces. A cylinder function has the property that it factors through a projection to a finite dimensional vector space. Such functions can be continuous and can be bounded, but (apart from the zero function) never vanish at infinity and never have compact support. Thus option 3 joins option 4 in the flowerbed.
As for option 2, I have no particular qualms about it except that it's not stable under partitions-of-unity. Assuming that I have such, then any continuous function can be written as a sum of bounded functions so when doing standard p-of-1 constructions I have to assume that my starting family is uniformly bounded (if that's the right term).
Well, I just remembered one qualm about option 2: if I go up the scale of differentiability then it gets increasingly hard to justify global bounds on the derivatives. I have a memory of John Roe telling me of some result that he'd proved which was to do with bounding all derivatives of a smooth function in some fashion. I don't recall the exact conditions, but the conclusion was that the only functions that satisfied them were trigonometric.
As others have said, if you are really only interested in (locally) compact spaces then the other options have meaning (functionally Hausdorff - points separated by functions - is assumed). But then the title of your question should have been: "Which is the correct ring of functions for a Locally Compact Hausdorff Space?".
I will take the question at face value, but not in the sense of justifying the definition.
A topological space is a convenient way of encoding, or perhaps better, organising, certain types of information. (Vague but true! I will give some instances. the data is sometimes `spatial' but more often than not, is not.)
Perhaps we should not think of spaces as 'god given' merely 'convenient', and there are variants that are more appropriate in various contexts.
A related question, coming from an old Shape Theorist (myself) is : when someone starts a theorem with 'Given a space $X$...', how is the space 'given'? As an algebraic topologist I sometimes need to use CW-complexes, but face the inconvenience that if I could give the CW structure precisely I could probably write down an algebraic model for its homotopy type precisely, and vice versa, so a good model is exactly the same as the one I started with. I hoped for more insight into what the space 'was' from my modelling. Giving the space is the end of the process, not the beginning. Strange. A space is a pseudo-visual way of thinking about 'data', which encodes important features, or at leastsome features that we can analyse, partially.
If someone gives me a compact subspace of $\mathbb{R}^n$, perhaps using some equations and inequalities, can I work out algebraic invariants of its homotopy type, rather than just its weak homotopy type? The answer will usually be no. Yet important properties of $C^*$ algebras on such a space, can sometimes be related to algebraic topological invariants of the homotopy type.
Spaces can arise as ways of encoding actual data as in topological data analysis, where there is a 'cloud' of data points and the practitioner is supposed to say something about the underlying space from which the data comes. There are finitely many data points, but no open sets given, they are for the data analyst to 'divine'.
Not all spatial data is conveniently modelled by spaces as such and directed spaces of various types have been proposed as models for changing data. Models for space-time are like this, but also models for concurrent systems.
Looking at finite topological spaces is again useful for encoding finite data (and I have rarely seen infinite amounts of data). For instance, relations between finite sets of data can be and are modelled in this way. Finite spaces give all homotopy types realisable by finite simplicial complexes. Finite spaces can be given precisely (provided they are not too big!) How do invariants of finite spaces appear in their structure? (Note the problem of infinite intersections does not arise here!!!)
At the other extreme, do we need points? Are locales not cleaner beasties and they can arise in lots of algebraic situations, again encoding algebraic information. Is a locale a space?
I repeat topological spaces are convenient, and in the examples you cite from algebraic geometry they happen to fit for good algebraic reasons. In other contexts they don't. Any Grothendieck topos looks like sheaves on a space, but the space involved will not usually be at all `nice' in the algebraic topological sense, so we use the topos and pretend it is a space, more or less.
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What about finite topological spaces? (A useful source of stuff on these is: Algebraic Topology of Finite Topological Spaces and Applications by Jonathan Barmak.) That area studies non-Hausdorff spaces most of the time and has strong links with CW-complexes via face posets but also via the link with posets has external contacts to combinatorics and to some of your other islands.
In another direction the use of topological spaces in Logic and Theoretical Computer Science should fit somewhere. One entry point is `Topology via Logic' by Steve Vickers. This fits near to some of your existing islands so will be linked to them by bridges (probably with tolls!). There is also a use of topological spaces within Modal Logic which again looks to be distinct to the others but linked.
Finally 'pathological' is not really definable except as meaning 'outside my current interests'! Pathology is in the eye of the beholder. Spaces such as compact Haudorff spaces have a decent algebraic topology if one uses strong shape theory. This approximates these spaces by CW spaces and transfers the well loved homotopy theory of those across using procategorical methods. Even general closed subsets of $\mathbb{R}^n$ which can look pathological can be explored. There are connections between their $C^*$-algebras and their strong shape, so linking the Banach space approaches with an extended CW-approach.
(I will stop there as that leads off into non-commutative spaces, and lots of other lovely areas, such as sheaves and toposes, but is getting to the limits of stuff I know at all well!)