1) Silverman's book mentioned above is surely a very good reference. For the first steps you can also consider Silverman-Tate "Rational points on elliptic curves". As you already use Hartshorne, I assume that you are familiar with basic notations in algebraic geometry and scheme theory. So you may wish to switch to references on general abelian varieties at some point. (Elliptic curves are abelian varieties of dimension $1$.) For abelian varieties I suggest Milne's article in the Storrs book "Arithmetic geometry" and the wonderful "prebook" on ablian varieties written by Moonen and van der Geer. To my knowledge this prebook is not published yet, but it can be downloaded on the homepage of Ben Moonen.

2) This is true, as mentioned above. An alternative definition: An elliptic curve is an abelian variety of dimension $1$. Here an abelian variety over $K$ is a (geometrically) integral proper group scheme over $K$.

3) I can give a few basic informations on Mordell-Weil groups: Let $K$ be a field and $E/K$ an elliptic curve. For $n$ coprime to $char(K)$ you have $E(\overline{K})[n]\cong ({\mathbb Z}/n)^2$, hence you know that $E(K)[n]$ is always isomorphic to a subgroup of $({\mathbb Z}/n)^2$. If $char(K)=p>0$, then there is an integer $f\in\{0, 1\}$ such that $E(\overline{K})[p^i]\cong ({\mathbb Z}/p^i)^f$ (and consequently $E(K)[p^i]$ is isomorphic to a subgroup of $({\mathbb Z}/p^i)^f$) for all $i\ge 1$.

If $K$ is finitely generated (over its prime field), then it is known that $E(K)$ is a finitely generated ${\mathbb Z}$-module by the so called Mordell-Weil-Lang-Neron theorem. (Cf. The article of Brian Conrad "Chows $K/k$-trace and $K/k$-image, and the Lang-Neron theorem (via schemes)".)

If $K$ is finite with $|K|=q$, then clearly $E(K)$ is finite. In addition to the information above, you then have the Hasse-Weil bound on the size of $E(K)$:
$$||E(K)|-q-1|\le 2\sqrt{q}.$$

4) I am not sure whether I interpret this question in the right way. But you can of course take your favorite elliptic curve $E$ over ${\mathbb F}_5$, given by an explicit Weierstrass equation, and and use a computer to make a list of the points in $E({\mathbb F}_5)$. (Just check which of the $31$ points in ${\mathbb P}_2({\mathbb F}_5)$ lie on $E$.)