For example, let's take a look at arcsin.
If I try to graph arcsin(x) in a graphing calculator. It turns out like this (Graph of arcsin(x) in GeoGebra). Shouldn't it look more like this (Graph of x=sin(y) in GeoGebra) since y=arcsin(x) is the same equation as x=sin(y)? Why does it stop like that? I comprehend that in order for an equation to be a function every point on the x-axis must have exactly one image on the y-axis, but wouldn't that just mean arcsin, arccos and arctan aren't functions?
Actually no, $x=\sin(y)$ is not equivalent to $y=\arcsin(x)$. Someone told you that $\arcsin$ is the inverse of $\sin$, but as you've noticed $\sin$ does not have an inverse. By definition $y=\arcsin(x)$ if and only if $x=\sin(y)$ and $-\pi/2\le y\le\pi/2$.