How many unique counting functions there are for strings of N bits?
Suppose that a set of functions $F_0,\ldots,F_{N-1}$ satisfies the constraints, generating the sequence $x_0 = 0^N, x_1, \ldots, x_{2^{N-1}-1}$. When does a function $F_N$ satisfy the constraints?

If $F_N$ works, then it generates a sequence of the form
$$ x_0\alpha_0, \ldots, x_{2^N-1}\alpha_{2^N-1}, \\x_0\alpha_{2^N}, \ldots, x_{2^N-1} \alpha_{2^{N+1}-1}. $$
The $\alpha_i$'s are given by the following rules: $\alpha_0 = 0$, and for $0 \leq i \leq 2^N-1$,
$$
\alpha_{i+1} = F_N(x_i\alpha_i), \\
\alpha_{2^N+i+1} = F_N(x_i\alpha_{2^N+i}).
$$
In order for the period to be $2^{N+1}$, we must have $\alpha_{i+1} \neq \alpha_{2^N+i+1}$. This means that there is a function $G_N\colon \{0,1\}^{N-1} \to \{0,1\}$ such that
$$ F_N(x\alpha) = G_N(x) \oplus \alpha. $$
It follows that for $0 \leq i \leq 2^N-1$, we have
$$
\alpha_{i+1} = F_N(x_i \alpha_i) = G_N(x_i) \oplus \alpha_i, \\
\alpha_{2^N+i+1} = F_N(x_i \alpha_{2^N+i}) = G_N(x_i) \oplus \alpha_{2^N+i}.
$$
In particular, this implies that
$$
\alpha_{2^N} = G_N(x_0) \oplus \cdots \oplus G_N(x_{2^N-1}).
$$
We must have $\alpha_{2^N} = 1$ in order to have a period of $2^{N+1}$, and this gives one constraint on $G_N$.
One can check that there are no other constraints, and so the number of possibilities for $F_N$ is $2^{2^N-1}$. In total, the number of possibilities is
$$
2^{2^0-1 + \cdots + 2^N-1} = 2^{2^{N+1}-N-2}.
$$
When $N=0,1,2$, this works out to be $2^0,2^1,2^4$. The corresponding sequences are:

For $N=0$, we have one sequence, $0, 1$.

For $N=1$, we have two sequences, $00, 10, 01, 11$ and $00, 11, 01, 10$.

For $N=2$, we have 16 sequences:
$$
000, 100, 010, 110, 001, 101, 011, 111 \\
000, 100, 010, 111, 001, 101, 011, 110 \\
000, 100, 011, 110, 001, 101, 010, 111 \\
000, 100, 011, 111, 001, 101, 010, 110 \\
000, 101, 010, 110, 001, 100, 011, 111 \\
000, 101, 010, 111, 001, 100, 011, 110 \\
000, 101, 011, 110, 001, 100, 010, 111 \\
000, 101, 011, 111, 001, 100, 010, 110 \\
000, 110, 010, 100, 001, 111, 011, 101 \\
000, 110, 010, 101, 001, 111, 011, 100 \\
000, 110, 011, 100, 001, 111, 010, 101 \\
000, 110, 011, 101, 001, 111, 010, 100 \\
000, 111, 010, 100, 001, 110, 011, 101 \\
000, 111, 010, 101, 001, 110, 011, 100 \\
000, 111, 011, 100, 001, 110, 010, 101 \\
000, 111, 011, 101, 001, 110, 010, 100 
$$