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?".
For a commutative ring $A$, $MaxSpec(A)$ is Hausdorff if and only if $A/JacobsonRadical(A)$ is a Gelfand ring (i.e. all equations of the form $(1-xb)\cdot (1-y(1-b))=0$, with $b$ from the ring, are solvable in that ring). $MaxSpec(A)$ is boolean if and only if every element of $A/JacobsonRadical(A)$ is a sum of a unit and an idempotent.
The Jacobson radical of $A$ is the intersection of all maximal ideals from $A$.
Both characterisations can be found in
N. Schwartz, M. Tressl; Elementary properties of minimal and maximal points in Zariski spectra. Journal of Algebra 323 (2010) 698-728.
This paper also characterises further properties of $MaxSpec(A)$ in terms of algebraic properties of the ring (see section 11 for an overview).
Best Answer
Peter Johnstone's book Stone Spaces (p. 144) proves that for any X, maximal ideals in $C(X)$ are the same as maximal ideals in $C_b(X)$ (bounded functions), i.e. the Stone-Cech compactification $\beta X$. Indeed, if I is a maximal ideal, let Z(I) be the set of all zero sets of elements of I; this is a filter on the lattice of all closed sets that are zero sets of functions. Then I is contained in J(Z(I)), the set of functions whose zero sets are in Z(I), so by maximality they are equal. But also, by maximality, Z(I) must be a maximal filter on the lattice of zero sets, and we get a bijection between maximal filters of zero sets and maximal ideals in $C(X)$. Now the exact same discussion applies to $C_b(X)$ to give a bijection between maximal filters of zero sets and maximal ideals of $C(X)$ (since the possible zero sets of bounded functions are the same as the possible zero sets of all functions). But the maximal ideals of $C_b(X)$ are just $\beta X$.
The difference between $C_b(X)$ and $C(X)$ is that for $C_b(X)$, the residue fields for all of these maximal ideals are just $C$, while for $C(X)$ you can get more exotic things. Indeed, if a maximal ideal in $C(X)$ has residue field $C$, then every function on X must automatically extend continuously to the corresponding point of $\beta X$. This can actually happen for noncompact X, e.g. the ordinal $\omega_1$.
Section IV.3 of Johnstone's book has a pretty thorough discussion of this stuff if you want more details.