Is it possible to find a context in which $A\to B$ and $\neg A\to\neg B$? I know that "if $A$ then $B$" does not mean "if $\neg A$ then $\neg B$" in the logical sense but in real life can it be possible to find a scenario that satisfies both statements and still perseveres the logic? For example, I can say if the car engine works, then the car moves. However, similarly, I can still say if there is a failure in the car engine, then the car does not move. Both statements are plausible to comprehend but not so at the logical sense, is it?
if A then B and if B then A ( also called A if and only if B (in short hand, A iff B)) implies not A then not B
When you declare something like : "A => B" , you are not specifically stating on A or B , you are declaring that the relation "=>" is true, no matter the trueness of A and B

Yes, it occurs that  "A => B" and "not A => not B". You may find a lot of examples, particularly when "A &lt;=> B". 

Indeed , $(A &lt;=&gt; B)  &lt;=&gt; (! A &lt;=&gt; ! B)$ and then it is an opportunity to say that we have both $(A =&gt; B)$  and $(! A =&gt; ! B)$ true.