Can every function be a composite to itself?
like we have $f:A\to B$ is $f \circ f$ always defined?
Can we say that if $f$ is a injection/surjection/bijection then so is $f\circ f$?
Also, how do we know if a composite between two functions is defined?
Let me take the standpoint of set theory, that a function is just a collection of ordered pairs with some special properties. (With the understanding that $f\colon A\to B$ means that the domain of $f$ is $A$ and its range is a subset of $B$.)
The composition of $f\circ f$ is indeed well-defined, and it will be a function, but it might be the empty function. The domain of $f\circ f$ will be, as it is easy to verify, $A\cap\operatorname{rng}(f)$, which may or may not be empty.
It is also quite easy to check that composing injective functions is again an injective function. However, when we say that a function is surjective, we make an appeal to an additional set, to $B$, so in order for the definition of "surjective" to make sense we need more than just $f$, rather we need $f\colon A\to B$, which will tell us non-trivial information about what $f$ is mapped onto, when we say it is a surjection.
Now $f\circ f$ isn't necessarily a function with domain $A$, and it's hard to say what exactly is its range, let alone when $f$ is some arbitrary function.
What is true, however, is that if $f\colon A\to B$ and $g\colon B\to C$ are surjective functions (and now we know that by surjective we mean that $\operatorname{rng}(f)=B$ and $\operatorname{rng}(g)=C$), then $g\circ f\colon A\to C$ is a surjective function (by which we mean that $\operatorname{rng}(g\circ f)=C$).