How many realizations are there then in a structured set of matrices yielding characteristic polynomial $(t+1)^4$? Let us consider a subset $S$ of $M_4(\mathbb R)$ which has following form
\begin{align*}
\begin{pmatrix}
0 & * & 0 & * \\
1 & * & 0 & * \\
0 & * & 0 & * \\
0 & * & 1 & *
\end{pmatrix},
\end{align*}
where $*$ can assume any real number. 
Is there a systemic way to determine how many realizations there are of $p(t) = (t+1)^4$ in $S$ up to Jordan forms? That is, I would like to know whether 
all Jordan blocks with diagonals to be $-1$'s are realizable in $S$? Obviously,
\begin{align*}
J_1 = \begin{pmatrix}
-1 & 0 & 0 & 0 \\
1 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 1 & -1
\end{pmatrix}, \quad J_2 = \begin{pmatrix}
-1 & 0 & 0 & 0 \\
1 & -1 & 0 & 0 \\
0 & 1 & -1 & 0 \\
0 & 0 & 1 & -1
\end{pmatrix}
\end{align*}
are realizable by choosing
\begin{align*}
A_1 = \begin{pmatrix}
0 & -1 & 0 & 0 \\
1 & -2 & 0 & 0 \\
0 & 0 & 0 & -1\\
0 & 0 & 1 & -2
\end{pmatrix}, \quad 
A_2 = \begin{pmatrix}
0 & 0 & 0 & -1 \\
1 & 0 & 0 & -4 \\
0 & 1 & 0 & -6\\
0 & 0 & 1 & -4
\end{pmatrix},
\end{align*}
For other cases, I am  essentially experimenting and try. But with no luck to get any result.
 A: 
The Jordan form of a matrix of your desired shape with characteristic polynomial $(1 + t)^4$ has nilpotent part of rank 2 or higher, and all such Jordan forms can be realized by such a matrix.

Split up the Jordan form $J$ into a diagonal part $J_d$ and a lower triangular nilpotent part $N$:
$$J = J_d + N,$$
where $J_d = \text{diag}(-1,-1,-1,-1)$. This is super handy, because we have the special case where $J_d$ commutes with every matrix. I am going to divide this up into a few cases depending on the rank of the matrix.
Case 1: $N = 0$. Then, it is not possible for $J$ to be similar to a matrix of your desired form; the only matrix similar to $J = J_d$ is $J_d$ itself, which cannot be represented in the form you seek.
Case 2: $N$ is a rank 1 matrix. A such representation is in this case also not possible: Let $M \in \mathbb{R}^{4 \times 4}$ be a matrix of your desired shape and $M = A \cdot J \cdot A^{-1}$ with some matrix $A \in GL_4(\mathbb{R})$.  This similarity would imply:
$$ M = J_d + A N  A^{-1}.$$
Since $N$ is of rank 1, so is $A N A^{-1}$, so it can be represented in the form 
$$ANA^{-1} = v \cdot  w^T$$
with vectors $v,w \in \mathbb{R}^4.$ Writing out the equality $M = J_d + v \cdot w^T$ yields an incompatible set of equations in the first and third column: Taking the equation for the $(1,1)$-,$(3,1)$- and $(3,3)$-entries for example yields
$$1 = v_1 w_1, \quad 
0 = v_1 w_3, \quad
1 = v_3 w_3.$$
From these follow $w_1 \neq 0, w_3 \neq 0$, and thus simultaneously $v_1 = 0$ and $v_1 = \frac{1}{w_1}$.
Case 3: $N$ is of rank 2 with one Jordan block of size 3 and one of size 1. For this case, consider for example the matrix
$$\begin{pmatrix} 
0 &1 &0 &-2 \\
1 &0 &0 &-2 \\
0 &2 &0 &-3 \\
0 &2 &1 &-4 \\ 
\end{pmatrix},$$
which is in your desired form and similar to a matrix with one Jordan block of size 3 and one of size 1. I give an explanation for how I found this example below. All other cases were already handled in the OP. $$\tag*{$\square$}$$

How I derived the last example: The case where $N$ possesses Jordan blocks of size 1 and 3 is equivalent to demanding $(M-J_d)^2 \neq 0$ but $(M-J_d)^3 = 0$.
Remember:
$$M - J_d =  \begin{pmatrix} 
1 &* &0 &* \\
1 &* &0 &* \\
0 &* &1 &* \\
0 &* &1 &* \\ 
\end{pmatrix}.$$
 Note that the matrix $M - J_d = ANA^{-1}$ is of rank 2, as $N$ is of rank 2, and $M - J_d$ already possesses two linearly independent vectors in column 1 and 3, so column 2 and 4 need to be linear combinations of those. As such, the most general form for $M - J_d$ is
$$M - J_d = \begin{pmatrix} 
1 &a &0 &c \\
1 &a &0 &c \\
0 &b &1 &d \\
0 &b &1 &d \\ 
\end{pmatrix}$$
with parameters $a,b,c,d \in \mathbb{R}$. By writing the equations down explicitly, one easily receives that $(M - J_d)^2 = 0$ is equivalent to the conditions $a = d = -1, b = c = 0$.
The equations for $(M - J_d)^3 = 0$ are a fair bit more convoluted (click here), but you can play around with these entries a bit, and for example demand that $d = -a - 2$, from which you then simply get the easy condition
$$(a + 1)^2 + bc = 0.$$
If this is fulfilled, $(M - J_d)^3$ vanishes, and now you just need to pick the parameters in a way that $(M - J_d)^2$ does not vanish. In the matrix example I gave above, I, for example, picked the parameters $a = 1,\, b = 2,\, c = -2,\, d = -3$, fulfilling the equation.
