Find all possible values of $a, p$ for which $x_{n+1} = x_n^2 + (1-2p)x_n + p^2$ converges, given $x_1 = a$ and $n\in\Bbb N$

Instead of viewing it as a map on $x_n$, try examining what happens to $x_n-p$.

From the second to last line of your display, it's easy to work out that $(x_{n+1}-p) = (x_n-p)^2 + (x_n-p).$

So the sequence $b_n=x_n-p$ (which converges exactly when $x_n$ does) is iterating under them map $t\mapsto t^2+t$. Now you have one function to understand not a class. It sounds like you have the tools necessary to verify this converges if the initial value $b_1$ is between -1 and 0.