Since Deane already answered your first question, let me address the other two.

If $\pi: E \to M$ is a bundle over a topological space $M$, you can define a sheaf on $M$ that associates to each open set $U \subseteq M$ the set of sections over it, i.e., maps $\sigma: U \to E$ such that $\pi \circ \sigma = \mathrm{id}_{U}$. Conversely, given a sheaf $\mathcal{F}$ on $M$ you can construct a topological space such that your $\mathcal{F}$ is its sheaf of sections. This Wikipedia page has some information on it. You will also be able to find information on any introductory book an algebraic geometry (e.g., Hartshorne).

Right inverses are called retractions, the same way left inverses are called sections. The name comes from topology too and that's how I think about them. Check an algebraic topology book for it (Spanier comes to mind, chapter 1 probably).

If $\pi: E \to M$ is a bundle over a topological space $M$, you can define a sheaf on $M$ that associates to each open set $U \subseteq M$ the set of sections over it, i.e., maps $\sigma: U \to E$ such that $\pi \circ \sigma = \mathrm{id}_{U}$. Conversely, given a sheaf $\mathcal{F}$ on $M$ you can construct a topological space such that your $\mathcal{F}$ is its sheaf of sections. This Wikipedia page has some information on it. You will also be able to find information on any introductory book an algebraic geometry (e.g., Hartshorne).