The order of the composition is important. The notation is read from left to right, so $g\circ f$ means apply the function $f$ first and the function $g$ second. The reason for this order is when you write write down the notation to plug $x$ into the function it is $(g\circ f)(x)$, and the function $f$ is applied to $x$ first.

Function composition only exists when the codomian of the first function is equal to the domain of the second function. (Though because you can always make the codomain bigger than needed, you could also make function composition work if the codomain of the first function is a subset of the domain of the second function.)

It is possible to preform function composition in both directions if the domain and the codomain of the same set. For example if $f:A\to A$ and $g:A\to A$ then both $g\circ f:A\to A$ and $f\circ g:A\to A$ are defined.

The identity function on a set $A$ is defined as the function ${\text{id}}_A:A\to A$ given by ${\text{id}}_A(x)=x$.

If $f:A\to B$ is a function then its composition with the identity function is itself:

Note the direction the identity function is applied. This is so that the domain/codomain line up correctly for function composition. (Also this means that if $A\ne B$ then $f\circ {\text{id}}_B$ and ${\text{id}}_A\circ f$ are undefined functions.)