Using Riemann-Roch Theorem to show every elliptic curve can be written as a plane cubic

Question

I've been studying how to show that every elliptic curve can be written as a plane cubic through the book of Joseph H. Silverman "Arithmetic Elliptic Curves", the proof of proposition III.3.1 (a) on page 64:

And the start of the proof:

My questions are the following

1. Why do we start by looking at the vector spaces $\mathscr{L}(n(O))$?
2. Why do we look at the particular cases of $n=2$ and $n=3$ (i.e. $\mathscr{L}(2(O))$ and $\mathscr{L}(3(O))$)
3. Given the general formula for a cubic, how and why do we restrict ourselves to the 7 functions $1,x,y,x^2,xy,y^2,x^3$?

Thank you for your time!