Algebraic independence of `Riemann-Roch' elements

Question

First of all, I'm not too sure on what terminology should be used in the title: the question deals with the vector spaces $$\mathcal{L(D)}=\{f\colon E\to\mathbb{C} \mid f\text{ is meromorphic}, (f)+D\geqslant0 \}$$ where $E$ is some elliptic curve and $D$ is some divisor on it. Bonus question: is there a standard name for this vector space? I can't find one in any of my books, or on Wikipedia. I know that it is also called $H^0(E,\mathcal{O}_E(D))$, but this is usually accompanied by a definition involving global sections or suchlike, which is a bit above my level at the moment.

But my main question comes from examining the case when $D=P$ is a point in $E$.

By Riemann-Roch we know that $$\ell(nD)=\begin{cases}1\quad\mbox{if }n=0 ,\\n\quad\mbox{for }n\geqslant1.\end{cases}$$ So $\ell(P)=1$, and so we can pick some $x\in\mathcal{L}(P)$ such that $\mathcal{L}(P)=\langle x\rangle$. Now $x^2\in\mathcal{L}(2P)$, and then we pick some $y\in\mathcal{L}(2P)$ such that $\mathcal{L}(2P)=\langle x^2, y\rangle$. Finally we do a similar thing to get that $\mathcal{L}(3P)=\langle x^3, xy, z\rangle$.

But this process, however simple, leaves me with a few questions:

1. How can we show that these sets (e.g. $\{x^2,y\}$) are linearly independent, and thus form a basis?

2. How can we then show that $x,y,z$ are all algebraically independent? EDIT: How can we show that $x,y$ are algebraically independent? On a problem sheet by Miles Reid he gives the hint

   Prove that $x^2,y$ defines a 2-to-1 map $E\to\mathbb{P}^1$ so that $x,y$ are algebraically independent. Prove that $y$ has a pole at $P$ of order $2$, and $z$ a pole of order $3$.

   First of all I'm not too sure what exactly he means by the first part, and I'm not too sure how to apply that fact even if I could prove it. Secondly, what is the second part of the hint getting at?

3. We know that there is a degree $6$ relation between $x,y,z$, and by a change of variables we can show that it takes a certain form ($z^2=y^3+ax^4y+bx^6$) provided that we can show that the relation must involve $y^3$ and $z^2$. How can we show this?

Guidance and hints would be appreciated as much as, if not more than, full answers!