For your final question, there are quite a number of applications of Pell's equation, such as (possibly in decreasing order of importance):
* Finding units in quadratic number fields (and proving results about them), as explained here.
* Solving some seemingly unrelated diophantine problems, such as finding numbers which are simultaneously square and triangle numbers (or more generally are simultaneously polygonal numbers of various types).
* Finding good rational approximations for $\sqrt{d}$. For example one of the solutions of $x^2-2y^2=1$ is $x = 577, y = 408$, and this gives the approximation $\sqrt{2} \simeq \frac{577}{408} = 1.4142156$.
* Convincing students that continued fractions are still worth studying these days.
* Proving the impossibility of solving a peculiar diophantine equation.