Well, the answer is well known of course. For a finitely generated module over a commutative normal Noetherian domain TFAE
- M is reflexive
- M is torsion-free and equals the intersection of its localizations at the codimension 1 primes
- M satisfies Serre's condition S2 and its support is Spec R.
- M is the dual of a f.g. module N
As you say, a finite projective module is the same as a locally free sheaf on Spec R. Similarly, a finite reflexive module is the same as the push forward of a locally free sheaf from an open subset U of Spec R whose complement has codimension $\ge2$.
So for an easy example take a Weil divisor D which is not Cartier, the associated divisorial sheaf (corresponding to an R-module of rank 1) is reflexive, not projective. Your example with a line on a quadratic cone is of this form.
This stuff is standard and used all the time in higher-dimensional algebraic geometry around the Minimal Model Program. For an old reference covering some of this, see e.g. Bourbaki, Chap.7 Algebre commutative, VII.
To your first question, "function on a space" $X$ usually means a morphism from $X$ to one of several "ground spaces" of choice, for example the reals if you work with smooth manifolds, Spec(A) if you work with schemes over a ring, etc. (This is a fairly selective use of the word "function" which used to confuse me.) A section $\gamma$ of a (some-kind-of) bundle $E\to X$ is thought of as a "generalized function" on $X$ by thinking of it as a funcion with "varying codomain", i.e. at each point $x\in X$, it takes value in the fibre
$E_x\to x$. If one is talking about locally free / locally trivial bundles, meaning $E$ is locally (over open sets
$U\subset X$) isomorphic to some product $U\times T$, then we can locally identify the fibres with $T$. Thus locally a section just looks like a function with codomain $T$, which is often required to be nice.
To your second question, I generally take the "right-inverse" or "pre-inverse" definition from category theory, because it relates back to others in the following precise way:
Say $\pi: Y\to X$ is a space over $X$ (intentionaly vague). The word "over" is used to activate the tradition of suppressing reference to the map $\pi$ and refering instead to the domain $Y$. For $U\subseteq X$ open, the notation $\Gamma(U,Y)$ denotes sections of the map $\pi$ over $U$, i.e. maps $U\to Y$ such that the composition $U \to Y\to X$ is the identity (thus necessarily landing back in $U$). It's not hard to see that
$\Gamma(-,Y)$ actually forms a sheaf of sets on $X$.
Conversely, given any sheaf of sets $F$ on a space $X$, one can form its espace étalé, a topological space over $X$, say $\pi: \acute{E}t(F) \to X$. Then for an open $U\subseteq X$, the elements of $F(U)$ correspond precisely to sections of the map $\pi$, which by the above notation is written $\Gamma(U,\acute{E}t(F)$. That is to say,
$F(-)\simeq\Gamma(-,\acute{E}t(F))$ as sheaves on $X$. This explains why people often refer to sheaf elements as "sections" of the sheaf.
Moreover, what we now denote by $\acute{E}t(F)$ actually used to be the definition of a sheaf, so people tend to identify the two and write $\Gamma(-,F)$ a instead of $\Gamma(-,\acute{E}t(F))$. This explains the otherwise bizarre tradition of writing $\Gamma(U,F)$ instead of the the more compact notation $F(U)$.
$\Big($Unfortunate linguistic warning: Many people incorrectly use the term "étale space". However, the French word "étalé" means "spread out", whereas "étale" (without the second accent) means "calm", and they were not intended to be used interchangeably in mathematics. This is unfortunate, because the espace étalé has very little to with with étale cohomology. More unfortunate is the annoying coincidence that when dealing with schemes the projection map from the espace étalé happens to be an étale morphism, because it is locally on its domain an isomorphism of schemes, a much stronger condition.$\Big)$
To your third question, I think the observation that $\Gamma(-,Y)$ forms a sheaf on $X$ gives a nice context in which to think of sections $X$ to $Y$: they "live in" the sheaf $\Gamma(-,Y)$ as its globally defined elements.
Best Answer
You need to refine the question to get better answers, but here are some thoughts:
1) Over a normal variety, you can think of line bundles as divisors, and "intersect" them.
2) A vector bundle can be represented by a reflexive sheaf, but being reflexive is a lot weaker.