If $C$ is any set of positive integers, and $g_C(n)$ is the number of permutations of $[n]$ whose cycle lengths are all in $C$, then $$G_C(x)=\sum_{n\ge 0}g_C(n)\frac{x^n}{n!}=\exp\left(\sum_{n\in C}\frac{x^n}{n}\right)$$ is the exponential generating function for the $g_C(n)$. (Rather than derive it, I’ve simply quoted this from Theorem 4.34 in Miklós Bóna, Introduction to Enumerative Combinatorics.)
In your case $C=\mathbb{Z}^+\setminus\{2\}$, so it’s $$\begin{align*}
G_C(x)&=\exp\left(\sum_{n\ge 1}\frac{x^n}n-\frac{x^2}2\right)\\
&=\exp\left(-\ln(1-x)-\frac{x^2}2\right)\\
&=\frac{e^{-x^2/2}}{1-x},
\end{align*}$$ and your generating function is correct. Then
$$\frac{e^{-x^2/2}}{1-x}=\left(\sum_{n\ge 0}x^n\right)\left(\sum_{n\ge 0}\frac{(-1)^nx^{2n}}{n!2^n}\right),$$ and
$$[x^n]\left(\sum_{n\ge 0}x^n\right)\left(\sum_{n\ge 0}\frac{(-1)^nx^{2n}}{n!2^n}\right)=\sum_{k=0}^{\lfloor n/2\rfloor}\frac{(-1)^k}{k!2^k}.$$
Recall, though, that the coefficient of $x^n$ in $G_C(x)$ is not $g_C(n)$, but rather $\dfrac{g_C(n)}{n!}$, so $$g_C(n)=n!\sum_{k=0}^{\lfloor n/2\rfloor}\frac{(-1)^k}{k!2^k}.$$
As a quick check, this yields $g_C(1)=1$, $g_C(2)=2\left(1-\frac12\right)=1$, $g_C(3)=6\left(1-\frac12\right)=3$, $g_C(4)=24\left(1-\frac12+\frac18\right)=15$, and $g_C(5)=120\left(1-\frac12+\frac18\right)=75$, all of which are in agreement with the OEIS values.
For your inclusion-exclusion argument, $c_k=0$ for $k>\lfloor n/2\rfloor$, and $c_0=n!$, so what you actually want is $$\sum_{k=0}^{\lfloor n/2\rfloor}(-1)^kc_k=\sum_{k=0}^{\lfloor n/2\rfloor}(-1)^k\frac{n!}{k!2^k},$$ which is exactly what we just got with generating functions.
Here is an argument completely different from joriki’s; it is also a complete solution.
A cycle is an odd permutation iff its length is even, so a permutation written as a product of disjoint cycles is even iff an even number of the factors are even cycles. Let $\pi=\sigma_1\dots\sigma_k$ be a permutation of $[n]$ written as a product of $k$ disjoint cycles. Then $n=|\,\sigma_1|+\dots+|\,\sigma_k|$, so the parity of $n$ is the same as the parity of the number of cycles of odd length. Thus, $\pi$ has an even number of cycles of even length iff $n$ and $k$ have the same parity, i.e., iff $(-1)^{n+k}=1$.
Let $e_n$ be the total number of cycles in even permutations of $[n]$, let $o_n$ be the total number of cycles in odd permutations of $[n]$, and let $d_n=e_n-o_n$. The total number of permutations of $[n]$ with $k$ cycles is given by $\left[n\atop k\right]$, the unsigned Stirling number of the first kind. Each of these $\left[n\atop k\right]$ permutations contributes $k$ cycles to $e_n$ if $(-1)^{n+k}=1$, and to $o_n$ if $(-1)^{n+k}=(-1)$. Thus, each contributes $(-1)^{n-k}k$ to $d_n$, and it follows that
$$d_n=\sum_k(-1)^{n+k}k\left[n\atop k\right]\;.$$
Now $(-1)^{n+k}\left[n\atop k\right]=(-1)^{n-k}\left[n\atop k\right]$ is the signed Stirling number of the first kind, for which we have the generating function $$\sum_k(-1)^{n+k}\left[n\atop k\right]x^k=x^{\underline{n}}\;.\tag{1}$$
(Here $x^{\underline{n}}=x(x-1)(x-2)\cdots(x-n+1)$ is the falling factorial, sometimes written $(x)_n$.)
Differentiate $(1)$ with respect to $x$ to obtain
$$\sum_k(-1)^{n+k}k\left[n\atop k\right]x^{k-1}=Dx^{\underline{n}}=(x-n+1)Dx^{\underline{n-1}}+x^{\underline{n-1}}\;,$$
where the last step is simply the product rule, since $x^{\underline{n}}=x^{\underline{n-1}}(x-n+1)$. But $$Dx^{\underline{n-1}}=\sum_k(-1)^{n-1+k}k\left[{n-1}\atop k\right]x^{k-1}\;,$$ so
$$\sum_k(-1)^{n+k}k\left[n\atop k\right]x^{k-1}=\sum_k(-1)^{n-1+k}k\left[{n-1}\atop k\right]x^{k-1}+x^{\underline{n-1}}\;.\tag{2}$$
Now evaluate $(2)$ at $x=1$ to get $$d_n=(2-n)d_{n-1}+[n=1]\;,\tag{3}$$ where the last term is an Iverson bracket. If we set $d_0=0$, $(3)$ yields $d_1=[1=1]=1$, which by direct calculation is the correct value: the only permutation of $[1]$ is the identity, which is even and has one cycle. It’s now a trivial induction to check that $d_n=(-1)^n(n-2)!$ for all $n\ge 1$: the induction step is
$$\begin{align*}d_{n+1}&=\Big(2-(n+1)\Big)d_n\\
&=(1-n)(-1)^n(n-2)!\\
&=(-1)^{n+1}(n-1)!\;.
\end{align*}$$
Added: It occurs to me that there’s a rather easy argument that does not use generating functions. Let $\sigma$ be any $k$-cycle formed from elements of $[n]$; then $\sigma$ is a factor in $(n-k)!$ permutations of $[n]$. Moreover, exactly half of these permutations are even unless $n-k$ is $0$ or $1$. Thus, $\sigma$ contributes to $d_n$ iff $k=n$ or $k=n-1$.
There are $(n-1)!$ $n$-cycles; they are even permutations iff $n$ is odd, so they contribute $(-1)^{n-1}(n-1)!$ to $d_n$.
There are $n(n-2)!$ $(n-1)$-cycles: there are $n$ ways to choose the element of $n$ that is not part of the $(n-1)$-cycle, and the other $n-1$ elements can be arranged in $(n-2)!$ distinct $(n-1)$-cycles. The resulting permutation of $[n]$ is even iff $n-1$ is odd, i.e., iff $n$ is even, so they contribute $(-1)^nn(n-2)!$ to $d_n$.
It follows that $$\begin{align*}d_n&=(-1)^nn(n-2)!+(-1)^{n-1}(n-1)!\\
&=(-1)^n\Big(n(n-2)!-(n-1)!\Big)\\
&=(-1)^n(n-2)!\Big(n-(n-1)\Big)\\
&=(-1)^n(n-2)!\;.
\end{align*}$$
Best Answer
Your answers are correct. Ignoring the alternating sign, in each summation the $j$ term is the number of ways to choose a set $F$ of $j$ odd numbers from $[n]$ and form a permutation of $[n]$ that fixes $F$ pointwise, so the alternating sum is exactly what the inclusion-exclusion argument requires: $j=0$ starts it off with all possible permutations of $[n]$, $j=1$ subtracts those with at least one specified fixed point, $j=2$ restores those that were subtracted twice, and so on.