Question: I asked 3 professors at my university and none gave me a clear cut answer, but instead merely told me qualities of this notation.
Here is what I understand so far from what they told me:   

1)Treat the top variable as as variable when finding the derivative  

2)Treat the bottom variable as a constant when finding derivative  

3)It it said "Find x with respect to y", but what exactly does that mean? What does it mean for something to be in respect to something else?

It seems like $\frac{dx}{dy}$ notation changes according to values in the problem. For instance, If $y = x^3 + 2x$ and $\frac{dx}{dt} = 5$, find $\frac{dy}{dt}$ when $x=2$. Why do the values of $\frac{dx}{dy}$ change in this problem and how do I solve this?

Answer: 
$\frac{d}{dx}y(x)$ denotes how $y$ varies according to $x$, and viceversa. It is only a metter of notation, even though there's a lot more to know.

Moreover, in your last example:
$$
\frac{dx}{dt}=5\Rightarrow x(t)=5t+x(0).
$$
So:
$$
\frac{dy}{dt}=\frac{dy}{dx}\cdot\frac{dx}{dt}=(3\cdot 2^2+2)\cdot 5.
$$
Here is an example:

Let $y=x^2.$ Then you want to find the derivative of $y$ with respect to $x$. Which means, you want to find $\frac{dy}{dx}$. In another form, $\frac{d}{dx}(x^2)$. Since $y=x^2$, then $\frac{d}{dx}(x^2)=\frac{d}{dx}(y)=\frac{dy}{dx}.$ Its a simplification of notation in a general sense.