Let's start by moving the summation terms all over to the right hand side.
$y(k) = -\sum_{i=1}^n a_i y(k-i) + \sum_{i=1}^n b_i u(k-i)$
Now, let's define $\mathbf{x_k} = \left(\begin{array}{c} y(k-1) \\ y(k-2) \\ \vdots \\ y(k-n) \end{array}\right)$, and $\mathbf{u_k} = \left(\begin{array}{c} u(k-1) \\ u(k-2) \\ \vdots \\ u(k-n) \end{array}\right)$.
Then, $\mathbf{x_{k+1}} = \mathbf{Ax_k}+\mathbf{Bu_k}$, where $\mathbf{A} = \rm{diag}(-a_1,-a_2,\ldots,-a_n)$, and $\mathbf{B} = \rm{diag}(b_1,b_2,\ldots,b_n)$.
Finally, $y(k) = \left(1\ 1\ \cdots\ 1\right)\mathbf{x_k}$.
Given a state space model of the following form,
$$
\dot{x} = A\,x + B\,u, \tag{1}
$$
$$
y = C\,x + D\,u. \tag{2}
$$
The openloop transfer function of this system can be found by taking the Laplace transform and assuming all initial conditions to be zero (such that $\mathcal{L}\{\dot{x}(t)\}$ can just be written as $s\,X(s)$). Doing this for equation $(1)$ yields,
$$
s\,X(s) = A\,X(s) + B\,U(s), \tag{3}
$$
which can be rewritten as,
$$
X(s) = (s\,I - A)^{-1} B\,U(s). \tag{4}
$$
Substituting this into equation $(2)$ and defining the openloop transfer function $G(s)$ as the ratio between output ($Y(s)$) and input ($U(s)$) yields,
$$
G(s) = C\,(s\,I - A)^{-1} B + D. \tag{5}
$$
In a normal block diagram representation the controller has as an input $r-y$, with $r$ the reference value you would like to have for $y$, and an output $u$, which would be the input to $G(s)$. For now $r$ can be set to zero, so the controller can be defined as the transfer function from $-y$ to $u$.
For an observer based controller ($L$ and $K$ such that $A-B\,K$ and $A-L\,C$ are Hurwitz) for a state space model we can write the following dynamics,
$$
u = -K\,\hat{x}, \tag{6}
$$
$$
\dot{x} = A\,x - B\,K\,\hat{x}, \tag{7}
$$
$$
\dot{\hat{x}} = A\,\hat{x} + B\,u + L(y - C\,\hat{x} - D\,u) = (A - B\,K - L\,C + L\,D\,K) \hat{x} + L\,y. \tag{8}
$$
Similar to equations $(1)$, $(2)$ and $(5)$, the transfer function of the controller $C(s)$, defined as the ratio of $U(s)$ and $-Y(s)$, can be found to be,
$$
C(s) = K\,(s\,I - A + B\,K + L\,C - L\,D\,K)^{-1} L. \tag{9}
$$
If you want to find the total openloop transfer function from "$-y$" to "$y$" you have to keep in mind that in general $G(s)$ and $C(s)$ are matrices of transfer functions, so the order of multiplication matters. Namely you first multiply the error ($r-y$) with the controller and then the plant, the openloop transfer function can be written as $G(s)\,C(s)$. The closedloop transfer function can then be found with,
$$
\frac{Y(s)}{R(s)} = (I + G(s)\,C(s))^{-1} G(s)\,C(s). \tag{10}
$$
It can also be found directly using equations $(2)$ and $(6)$, and the closedloop state space model dynamics,
$$
\begin{bmatrix}
\dot{x} \\ \dot{\hat{x}}
\end{bmatrix} = \begin{bmatrix}
A & -B\,K \\
L\,C & A - B\,K - L\,C
\end{bmatrix} \begin{bmatrix}
x \\ \hat{x}
\end{bmatrix} + \begin{bmatrix}
0 \\ -L
\end{bmatrix} r, \tag{11}
$$
$$
\frac{Y(s)}{R(s)} = \begin{bmatrix}
C & -D\,K
\end{bmatrix} \begin{bmatrix}
s\,I - A & B\,K \\
-L\,C & s\,I - A + B\,K + L\,C
\end{bmatrix}^{-1} \begin{bmatrix}
0 \\ -L
\end{bmatrix}. \tag{12}
$$
Best Answer
For the series state space model you just have to link the output of the first system to the input of the second system, so $u_2 = y_1 = C_1\,x_1 + D_1\,u_1$. By defining the new state vector as $x = \begin{bmatrix}x_1^\top & x_2^\top\end{bmatrix}^\top$, input as $u = u_1$ and output as $y=y_2$ yields the following combined system
\begin{align} \dot{x} &= \begin{bmatrix} A_1 & 0 \\ B_2\,C_1 & A_2 \end{bmatrix} x + \begin{bmatrix} B_1 \\ B_2\,D_1 \end{bmatrix} u \\ y &= \begin{bmatrix} D_2\,C_1 & C_2 \end{bmatrix} x + D_2\,D_1\,u. \end{align}
For the parallel state space model you just have to give both systems the same input and add the output of both systems for the final output, so $u_1 = u_2 = u$ and $y = y_1 + y_2$. By again defining the new state vector as $x = \begin{bmatrix}x_1^\top & x_2^\top\end{bmatrix}^\top$, input as $u = u_1 = u_2$ and output as $y = y_1 + y_2$ yields the following combined system
\begin{align} \dot{x} &= \begin{bmatrix} A_1 & 0 \\ 0 & A_2 \end{bmatrix} x + \begin{bmatrix} B_1 \\ B_2 \end{bmatrix} u \\ y &= \begin{bmatrix} C_1 & C_2 \end{bmatrix} x + \left(D_2 + D_1\right) u. \end{align}
For the feedback state space model I will just consider one state space model, since the equivalent state space model for $G_1\,G_2$ can be obtained using the series state space model. For feedback the input to the starting state space model is $u = r \mp y = r \mp (C\,x + D\,u)$. I will try to keep it as general as possible and also allow MIMO systems, so solving for $u$ gives $u = (I \pm D)^{-1} (r \mp C\,x)$, which yields the following simplified closed loop system
\begin{align} \dot{x} &= \left(A \mp B\,(I \pm D)^{-1} C\right) x + B\,(I \pm D)^{-1} r \\ y &= (I \pm D)^{-1} C\,x + D\,(I \pm D)^{-1} r, \end{align}
from which it can be seen that $I \pm D$ has to be invertible.
Of course just like any other state space model you can apply any similarity transformation you like, so these representations are not unique. Also these resulting state space models might not be minimal (controllable and observable) even if the starting model are. For example when considering the parallel systems and $G_1=G_2$.
More different combinations of state space models and additional gain matrices can also be found in Duke, Eugene L. "Combining and connecting linear, multi-input, multi-output subsystem models." (1986).