Perhaps there is something relating odd ordered partitions with even ordered partitions?
There is indeed. Let's try to construct an involution $T_n$, mapping odd ordered partitions of $n$-element set to even and vice versa:
if partition has part $\{n\}$, move $n$ into previous part; otherwise move $n$ into new separate part.
Example: $(\{1,2\},\{\mathbf{5}\},\{3,4\})\leftrightarrow(\{1,2,\mathbf{5}\},\{3,4\})$.
This involution is not defined on partitions of the form $(\{n\},\ldots)$, but for these partitions one can use previous involution $T_{n-1}$ and so on.
Example: $(\{5\},\{4\},\{1,2\},\{\mathbf{3}\})\leftrightarrow(\{5\},\{4\},\{1,2,\mathbf{3}\})$.
In the end only partition without pair will be $(\{n\},\{n-1\},\ldots,\{1\})$. So our (recursively defined) involution gives a bijective proof of $\sum_{\text{k is even}}k!{n \brace k}=\sum_{\text{k is odd}}k!{n \brace k}\pm1$ (cf. 1, 2).
Upd. As for the second identity, the involution $T_n$ is already defined on all cyclically ordered partitions, so $\sum_{\text{k is even}}(k-1)!{n \brace k}=\sum_{\text{k is odd}}(k-1)!{n \brace k}$.
P.S. I can't resist adding that $k!{n \brace k}$ is the number of $(n-k)$-dimensional faces of an $n$-dimensional convex polytope, permutohedron (the convex hull of all vectors formed by permuting the coordinates of the vector $(0,1,2,\ldots,n)$). So $\sum(-1)^{n-k}k!{n \brace k}=1$ since it's the Euler characteristic of a convex polytope.
One way to solve it: first put all $nk$ items in order: there are $(nk)!$ ways. Now chop them into blocks of $k$ and you have a partition. Each block can be reordered in $k!$ ways, then the blocks can be put in order in $n!$ ways. In all, we have $\frac{(nk)!}{(k!)^nn!}$ ways to make the partition.
Another way is to say there are ${nk \choose k}$ ways to get the first partition, ${(n-1)k \choose k}$ to get the second, and so on. Again you can choose the partitions in $n!$ orders. This gets you the same answer.
Best Answer
These are the ordered Bell numbers (or Fubini numbers)
$$a(n)=\sum_{k=0}^nk!{n\brace k}\;;$$
asymptotically they satisfy
$$a(n)\approx\frac{n!}{2(\ln 2)^{n+1}}\;.$$
The sequence of these numbers is OEIS A000670, where you’ll find many references and much information; there don’t seem to be any nice closed forms. You’re interested specifically in $a(n)$ when $n$ is a power of $2$; it does not appear to me that this helps.