How can I use the relationship between the Fibonacci numbers and the EA to fill squares? 
For this first question, I know how to apply the Euclidean algorithm and if I do, I get that the gcd is 1. I found this theorem online, thinking it might be able to piece together the relationship between the Euclidean algorithm and the Fibonacci numbers. 

When it asks "What squares could you use to fill a 6765 x 4181 rectangle," I don't know what this means. 
What are the next steps in solving this problem? And is there any deeper relationship between the Fibonacci numbers and EA?
For the second question, I am assuming it is something similar as well. 
HDYK just means "how do you know", I'm assuming.
 A: I will address your first question.
While I think that the question is poorly phrased, I think I know what they mean by "What squares could you use to fill up a 6765$\times$4181 rectangle". I will illustrate this on a scaled-down version of the problem.
We have that $F_6=8$ and $F_5=5$. Let's fill up a $8\times5$ rectangle with squares.

One can see that we can fill up the rectangle perfectly with squares whose sides' lengths are all Fibonacci numbers up to $F_5$.
Why does this work for every pair of consecutive Fibonacci numbers?
Imagine that you are starting from one corner (e.g. the top right one as in the image) of the rectangle by placing a $1\times1$ square in it. Then, along the smaller side of the rectangle, place another $1\times1$ square. The two squares form another rectangle. 
Now place another square ($2\times2$) next to the longer side of that rectangle. What results is yet another rectangle. Now just repeat the process until you use up all the Fibonacci numbers up to $F_6$.
Notice that by adding new squares, we will always get a new rectangle. What I mean by this is that the resulting figure will never be "jagged". This follows from the way that Fibonacci numbers are defined.
I think that this is what the question was going for, although it was not very well phrased, since there are many ways to fill up a rectangle with squares.

EDIT: The second part of the question
As Jair Taylor suggested in a comment, the number $F_{2018}/F_{2017}$ is rational and therefore its decimal representation will either have a finite number of digits or it will be periodic. This means that the decimal digits will contain repeating "chunks" of numbers. For example, in the decimal representation of the number $70/111$, the sequence "$630$" keeps repeating:
$$70/111=0.\underbrace{630}\underbrace{630}\underbrace{630}...$$
Assuming that $\phi$ is the golden ratio, $\phi=\frac{1+\sqrt5}{2}$, its decimal representation will be unpredictable, because $\phi$ is an irrational number.
Looking at the pictures you provided, one can see that the first one is much more "chaotic" then the second one and one can hardly find any repeating patterns, whereas the second one obviously contains repeating patterns. This gives us good reason to conclude that the first image represents $\phi$ and the second one represents $F_{2018}/F_{2017}$.
