How should I think about homogeneous coordinates?

It's quite easy to make sense of ordinary coordinates on a space $X$: it is just an open embedding of an open subset of $X$ in some vector space. Along the same lines, one could say that homogeneous coordinates on $X$ is an embedding of $X$ into projective space, but I don't find this as helpful, since I don't feel I completely understand what is happening even in that case. (The locally ringed space formalism is technically convenient but I think what I am missing is the global picture.)

So, fix a base field $k$, which will be algebraically closed (or even $\mathbb{C}$ if that is particularly helpful). Let $C$ be $k^{n+1} \setminus \{ \vec{0} \}$ as a quasi-affine variety and let $\pi : C \to \mathbb{P}^n$ be the standard projection. The correspondence between points of $\mathbb{P}^n$ and homogeneous coordinates thought of as points in $k^{n+1}$ is then tabulated as $\mathbb{P}^n \leftarrow C \hookrightarrow k^{n+1}$ in the category of varieties, but this correspondence is obviously not a functional relation.

Question 1. It is well-known that $\pi : C \to \mathbb{P}^n$ does not have any global sections if $n \ge 1$, so it is not possible to make the relation functional in a uniform way. But $\pi$ does have local sections over dense open sets. How should these be thought of? It does not appear to have any geometrical significance.

Also, it is easy to make sense of morphisms $g : \mathbb{P}^n \to Y$ using homogeneous coordinates: just take the composition $g \circ \pi$. But how do we make sense of morphisms $f : X \to \mathbb{P}^n$? Because homogeneous coordinates are not functional on $\mathbb{P}^n$ there is no obvious way of pulling them back along $f$ to get a coordinate expression for $f$ (i.e. a lift of $f$ through $\pi$).

Question 2. On the other hand, $\pi : C \to \mathbb{P}^n$ is a principal $\mathbb{G}_m$-bundle on $\mathbb{P}^n$, so we can pull that back along $f : X \to \mathbb{P}^n$ and hopefully get a principal $\mathbb{G}_m$-bundle on $X$. What is the significance of this, and does it have any connection with finding coordinate expressions for $f$?

More generally, how can we define morphisms $f : X \to \mathbb{P}^n$? After all, a morphism $X \to \mathbb{A}^n$ is the same thing as a family of $n$ regular functions on $X$, but as I understand it, there are morphisms $f : X \to \mathbb{P}^n$ that do not lift through $\pi : C \to \mathbb{P}^n$.

Question 3. Is there some collection of data on $X$, definable without reference to $\mathbb{P}^n$, such that there is a natural bijection between such data on $X$ and morphisms $X \to \mathbb{P}^n$? In other words, is there a nice interpretation of the presheaf represented by $\mathbb{P}^n$?

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It seems like the answer to your last question is: very ample invertible sheaves globally generated by $n+1$ sections (up to a common constant multiple). –  Andrew Feb 6 at 14:08
@Andrew If I recall correctly, such a thing defines an embedding into $\mathbb{P}^n$. I'm only asking for morphisms $X \to \mathbb{P}^n$. I've since convinced myself that these should be the same thing as line bundles over $X$ equipped with a chosen embedding into the trivial bundle $X \times k^{n+1}$. –  Zhen Lin Feb 6 at 14:17
Whoops, of course you are correct -- I shouldn't have added the adjective "very ample". –  Andrew Feb 6 at 14:41