Recursive function that outputs its own code

You want $\phi_c$ to be "the function which outputs $c$". Hence, you want $\phi_{f(c)}$ also to be a function that outputs $c$. The difference is that you can control $\phi_{f(c)}$ since you control $f(x)$.

So you want to define $f(x)$ to be the code of the constant function $x$ (i.e. returns $x$ for every input). Now Kleene's theorem gives you the following: There exists $c$ such that the function coded by $c$ is exactly the constant function $c$.

Let $T_i$ be a Turing Machine with Godel number $i$ and $\phi_i$ the function computed by $T_i$.

We wish to show there exists a number $n_0$ such that $T_{n_0}$ prints its own coding $n_0$. Let $\phi_0, \phi_1, ...$ be any enumeration (Godel numbering) of all computable functions and let $h$ be any totally computable function. Then, by the fixed point theorem (as you have suspected) there exists a number $i_0$ such that $\phi_{i_0} = \phi_{h(i_0)}$, i.e. $i_0$ is a fixed point for $h$.

Let us now define the function $h$ to be the function such that for each natural number $n$, we have that $h(n)$ is equal to a machine that prints $n$ and halts, i.e., such that $\phi_{h(n)} = n$. Let $n_0$ be a fixed point for $h$. Then $\phi_{n_0} = \phi_{h(n_0)} = n_0$.