Proof that no DES key makes encryption identity
To throw something, here are simple proofs for 2-4 rounds, applicable to almost any Feistel Network. Some notation: $F_i$ denotes $i$-th Feistel round without swaps, $f_i$ denotes the Feistel function at $i$-th round, $S$ denotes the swap.
Since the initial permutation IP and the final permutation FP are inverses of each other, we can strip them away: for any bijection $P$:
$$P^{-1} \circ G \circ P = Id ~~~\Leftrightarrow~~~ G = Id.$$
We will use this trick later.
(2 rounds)

Consider a 2-round Feistel network (like in DES, with no swaps in the beginning and in the end), denote it by $G$. We want to check if it holds that
$$\forall a,b \in \mathbb{F}_{32} ~~~ G(a,b) = a,b.$$
The right output half is equal to $a \oplus f_1(b)$. This can not always equal $b$ for any $a$ (consider two distinct $a$).
(3 rounds)

Consider a 3-round Feistel network. The right half in the middle is equal to $a \oplus f_1(b)$ (from the input) and to $a \oplus f_3(b)$ (from the output). It follows that $f_1(b) = f_3(b)$ for all $b$. We can now use the trick again and remove symmetrically the first and the last round and the two middle swaps. We end up with $F_2(a,b) := (a \oplus f(b), b)$. Due to the nonlinearity of $f$ in DES, there exist no keys such that $f_2$ (and so $F_2$) is an identity (the key is only xored at the input of $f$).
(4 rounds)

Note that $(S\circ F_1)(a, b) = (b, *)$, and $(S \circ F_4^{-1})(a, b) = (b, *)$, where $*$ denotes a wildcard. So it sufficient to show that there are no 2-round FN that for all $b,x\in \mathbb{F}_{2^{32}}$ always map $(b, x)$ to $(b, y)$ for some $y \in \mathbb{F}_{2^{32}}$. For the two middle rounds we get 2 equations:
$$
b \oplus f_2(x) = y,
$$
$$
b \oplus f_3(y) = x.
$$
If in the input we change $b$ to $b\oplus \delta$ and don't change $x$, then $y$ has to change to $y \oplus \delta$ (from the first equation). From the second equation, we must have
$$b \oplus \delta \oplus f_3(y\oplus \delta) = x,$$
and so
$$f_3(y \oplus \delta) \oplus f_3(y) = \delta.$$
Since this must hold for all $y$, we obtain that $f_3$ must be a XOR with a constant, which obviously does not hold for DES rounds.