Schemes play an enormous role in all the modern theory of elliptic curves, and have done so ever since Mazur and Tate proved their theorem that no elliptic curve over $\mathbb Q$ can have a 13-torsion point defined over $\mathbb Q$.
For some additional explanation, you could look at this answer. But bear in mind that theorems on the classification of torsion, while fantastic, are just a tiny part of the theory of elliptic curves, and a tiny part of how schemes are involved. One of the most important theorems about elliptic curves is the modularity theorem, proved by Wiles, Taylor, et. al. twenty or so years ago, which implies FLT. These arguments also depend heavily on modern algebraic geometry.
Also, the proof of the Sato--Tate conjecture.
Also, all current progress on the BSD conjecture.
The underlying point is the the theory of elliptic curves is one of the central topics in modern number theory, and the methods of scheme-theoretic alg. geom. are among the central tools of modern number theory. So certainly they are applied to the theory of elliptic curves
On the other hand, you won't find it so easy to synthesize your reading on the arithmetic of elliptic curves and your reading of scheme theory. For example, even Silverman's (first) book, which is quite a bit more advanced than Silverman--Tate, doesn't use schemes. Some of the arguments can be clarified by using schemes, but it takes a bit of sophistication to see how to do this, or even where such clarification is possible or useful.
Hartshorne has a discussion of elliptic curves in Ch. IV, but it doesn't touch on the number theoretic aspects of the theory; indeed, Hartshorne's book doesn't make it at all clear how scheme-theoretic techniques are to be applied in number theory.
With my own students, one exercise I give them to get them to see how make scheme-theoretic arguments and use them to study elliptic curves is the following:
Let $E$ be an elliptic curve over $\mathbb Q$ with good or multiplicative reduction at $p$; then prove that reduction mod $p$ map from the endomorphisms of $E$ over $\overline{\mathbb Q}$ to the endomorphisms of the reduction of $E$ mod $p$ over $\overline{\mathbb F}$ is injective.
The proof isn't that difficult, but requires some amount of sophistication to discover, if you haven't seen this sort of thing before.
Finally:
None of the results on torsion on elliptic curves over $\mathbb Q$, or modularity, or Sato--Tate, or BSD, will be accessible to you in the time-frame of your masters (I would guess); any one of them takes an enormous amount of time and effort to learn (a strong Ph.D. student working on elliptic curves might typically learn some aspects of one of them over the entirety of their time as a student). I don't mean to be discouraging --- I just want to say that it will take time, patience, and also a good advisor, if you want to learn how schemes are applied to the theory of elliptic curves, or any other part of modern number theory.
If you define it to be a projective curve given by a degree 3 polynomial, you just need a formula linking the degree of a plane curve with the genus. This you can find:
http://en.wikipedia.org/wiki/Genus%E2%80%93degree_formula
(By smoothness, which you can check by hand, it doesn't matter if you want to refer to arithmetic or geometric genus).
It's a bit more natural to define it to be a (smooth projective) genus one curve with a marked point, and then prove every such object can be given an equation like the one you wrote, but it can be easier to define everything in terms of equations.
Best Answer
The group law on an elliptic curve was not discovered in a vacuum. It came up in the context of abelian integrals.
Let $y^2 = f(x)$, where $f(x)$ is a cubic in $x$ be an elliptic curve; call it $E$.
Elliptic integrals are integrals of the form $$\int_{a}^x dx/y = \int_a^x \frac{d x}{\sqrt{ f(x)}}.$$ (Here $a$ is some fixed base-point.) They come up (in a slightly transformed manner) when computing the arclength of an ellipse (whence their name).
If was realized at some point (in the 1600s or 1700s) (at least in special cases) that if you apply certain substitutions to $x$, you can double the value of the integral, or that if you apply certain substitusions of the form $x = \phi(x_1,x_2)$, the integral you compute is the sum of the individual integrals for $x_1$ and $x_2$.
Real understanding came from the work of Abel and Jacobi. (Unfortunately I don't know the precise history or attributions here.)
What they realized (in modern terms) is that, if we fix the base-point $a$ and let $b$ vary, then the elliptic integral is giving a multivalued map from the elliptic curve $E$ to $\mathbb C$, and that the formula $\phi(x_1,x_2)$ mentioned above shows us a way to add points on the elliptic curve, so that this map is a (multi-valued) group homomorphism.
Taking the inverse of this multi-valued map gives a single-valued map (which is how we are more used to thinking about it) $$\mathbb C \to E,$$ which is a homomorphism when we give $\mathbb C$ its addivite structure and $E$ the group law coming from $(x_1,x_2)\mapsto \phi(x_1,x_2)$. The kernel of this map turns out to be a lattice $\Lambda$, so that we get an isomorphism $\mathbb C/\Lambda \cong E.$
The formula $\phi(x_1,x_2)$ turns out to be precisely the formula describing addition on the elliptic curve via chords and tangents, and there are lots of theoretical explanations for it, as you can find on the linked MO page.
For a higher genus curve $C$, it turns out that there is not just the one holomorphic differential $dx/y$, but $g$ linearly independent such (if the curve has genus $g$), say $\omega_1, \ldots,\omega_g$. Furthermore, if we fix a particular differential $\omega_i$, then there is no formula $\phi(x_1,x_2)$ such that the sum of the integrals of $\omega$ for $x_1$ and $x_2$ is equal to the integral of $\omega$ for $\phi(x_1,x_2)$.
However, what Abel and Jacobi found is that, if we consider the map $$(x_1,\ldots,x_g) \mapsto (\sum_{i = 1}^g \int_a^{x_i} \omega_1, \ldots,\sum_{i = 1}^g \int_a^{x_i} \omega_g),$$ which gives a multi-valued map $$Sym^g C \to \mathbb C^g$$ (here $Sym^g C$ denotes the $g$th symmetric power of $C$, so it is the product of $g$ copies of $C$, modulo the action of the symmetric group on the $g$ factors), then we can find a formula $\phi(x_{1,1},\ldots,x_{1,g},x_{2,1},\ldots,x_{2,g})$ such that $\phi$ defines a group law on $Sym^g C$ (at least generically) and such that this map is a homomorphism.
Again, this map becomes well-defined and (generically) single valued if we quotient out the target by an appropriate lattice $\Lambda$ (the period lattice), to get a birational map $$Sym^g C \to \mathbb C^g /\Lambda.$$
The target here is called the Jacobian of $C$, and can be identified with $Pic^0(C)$.
In summary, to generalize the addition law on an elliptic curve to higher genus curves, you have to consider unordererd $g$-tuples of points (where $g$ is the genus), and add those (not just individual points).
(The relationship with $Pic^0(C)$ is that if $x_1,\ldots,x_g$ are $g$ points on $C$, then $x_1 + \ldots + x_g - g a$ is a degree zero divisor on $C$, and every degree zero divisor is linearly equivalent to a divisor of this form, and, generically, to a unique such divisor.)