Geometrical meaning of dx, dy and dy/dx? I am also confused about whether these are symbols or have some meaning of their own. 
PS- I know dy/dx geometrically represents the slope. But, I've come across dx/dy to make problems easier. What does dx/dy mean?
You should be very careful, those are merely notations.

$\frac{d\,y(x)}{d\,x}$ is the derivative of a variable $y$ with respect to $x$. It represents how much $y$ varies for small variations of $x$. If you draw the curve of $y(x)$, the derivative will be the slope, as you said.

The opposite works as well, if one can define $x$ as a function of $y$ (which is the same as inverting the function $y(x)$ - we must consider a subset of the image where it is injective to do so), you can differentiate it too: $\frac{d\,x(y)}{d\,y}$

But be careful! It is just a notation, it is not the division of $dx$ and $dy$, it is a limit as you can see here.

$\boxed{\frac{d\,y(x)}{d\,x} = \lim\limits_{h\rightarrow0} \frac{y(x+h)-y(x)}{h}}$

Here is some discussion about the meaning of $dx$ alone, pay attention to the commentary at the end: "Since I first posted this paper, two different people have emailed me to tell me that Real Mathematicians don't do this. Playing with dx in the ways described in this paper is apparently one of those smarmy tricks that physicists use to give headaches to mathematicians. "
In the terms dx and dy, the d is for delta or "change in". So they represent the change in y and the change in x as a function, usually in terms of each other but sometimes another parameter. So dy/dx as you said is the slope, or change in x divided by the change in y, dy/dx is simply the inverse slope.

The different between dy and ∆y or dx and ∆x is that dy is a function that can be solved at any point to give the change in y at that point in relation to another variable, where as ∆y is a numerical value representing the difference in y between two points.