To answer the second part of your question. there is a conjecture of Serge Lang which says that the number of integer points (on a minimal Weierstrass equation) is bounded solely in terms of the rank. The conjecture is known to be true if one assumes the $ABC$-conjecture, in the stronger form
$$ \#E(\mathbb Z) \le C^{1+\operatorname{rank}E(\mathbb Q)} $$
for an absolute constant $C$. So a sequence of curves with $\#E_i(\mathbb Z)\to\infty$ would (conjecturally) give a sequence with unbounded rank. However, there are those who believe that there should be an absolute upper bound for $\#E(\mathbb Z)$, and recently some who believe that there should be an absolute upper bound for $\operatorname{rank}E(\mathbb Q)$. Anyway, lots of interesting open questions.

For the first part of your question, there are many natural problems that end up coming down to integer points on elliptic curves. Just as there are many natural problems that come down to rational points on curves of genus $g\ge2$. But I'm not sure that's the right answer to your question. For me, it comes down to (1) integers and their relationships to one another are interesting, (2) the basic operations on the integers are addition and multiplication, so the basic functions are polynomial functions, (3) hence Diophantine equations (polynomial equations to be solved in integers) are intrinsically interesting. If you'll grant that, then people eventually realized that the "right" way to classify equations of the form $f(x,y)=0$ was by genus, the first case ($g=0$) is now well understood, and the second case ($g=1$) still presents challenges.