Are there any known examples of groups that virtually split that don't have a codimension-1 subgroup? Are there any known examples of groups that virtually split that don't have a codimension-1 subgroup?
 A: No. For a finitely generated group, the property "have no codimension 1 subgroup" is called Property FW. Property FW passes to finite index subgroups, so excludes virtual splittings. 
Actually for an infinitely generated group, the existence of a "codimension 1 subgroup", the transitive version of Property FW, is still discarded by virtual splittings, but there are maybe a few details to check. I'll assume in the absence of context that you're interested by finitely generated groups.
Edit: here are missing references. That Property FW passes to finite index subgroups is Proposition 5.B.1 in this arxiv paper. The non-existence of a "codimension 1 subgroup" is called Property FW' (and equivalent to Property FW for finitely generated groups, but not in general). That is passes to finite index subgroups is obtained by a similar argument. The whole point is that when one induces an action of a finite index subgroup to the whole group (see §4.F in the above reference), this maps a transitive action to an action with finitely many orbits. I don't use CAT(0) cube complexes for this.
A: In addition to Yves' answer, here are a few details. So the statement is:
Theorem: Let $G$ be a (not necessarily finitely-generated) group and $H \subset G$ a finite-index subgroup. If $H$ contains a codimension-one subgroup, then so does $G$.
The first step is Niblo and Roller's theorem: A group contains a codimension-one subgroup if and only if its acts fixed-point freely on a connected cube with a single orbit of hyperplanes (Groups acting on cubes and Kazhdan's property (T), 1998). The second step is notice that an action of the entire group on a CAT(0) cube complex can be constructed from such an action of a finite-index subgroup. I think that the idea of the construction goes back to Serre's article Cohomologie des groupes discrets (1971).
Proposition: Let $G$ be a group, $H \subset G$ an finite-index subgroup. Suppose that $H$ acts fixed-point freely on some CAT(0) cube complex $X$. Then $G$ acts fixed-point freely on $X^{[G:H]}$.
Proof. Consider the set
$$\mathrm{Hom}_H(G,X) = \{ f : G  \to X \mid \forall h \in H, \forall g \in G, \ f(hg)=h \cdot f(g) \}.$$
$G$ acts on $\mathrm{Hom}_H(G,X)$ via:
$$g \cdot f : x \mapsto f(g^{-1}x).$$
Fixing some representatives $G=Ha_1 \sqcup \cdots \sqcup Ha_r$, the map
$$\varphi(a_1, \ldots,a_r) : \left\{ \begin{array}{ccc} \mathrm{Hom}_H(G,X) & \to & \prod\limits_{i=1}^r X=X^{[G:H]} \\ f & \mapsto & (f(a_1), \ldots, f(a_r)) \end{array} \right.$$
is a bijection. Notice that, if $b_1, \ldots, b_r$ are other representatives, then
$$\varphi(b_1, \ldots, b_r) \varphi(a_1, \ldots, a_r)^{-1} : (x_1, \ldots, x_r) \mapsto (h_1x_{\sigma(1)}, \ldots, h_r x_{\sigma(r)}),$$
where $\sigma : \{1, \ldots, r \} \to \{1,\ldots,r\}$ is a permutation and $h_1, \ldots, h_r$ elements of $H$ such that $b_i=h_ia_{\sigma(i)}$ for every $1 \leq i \leq r$. Consequently, if we endow $X^{[G:H]}$ with its natural cubical structure, then then the previous map is an isomorphism. 
The conclusion is that, thanks to $\varphi(a_1, \ldots, a_r)$, you can transfer the action $G \curvearrowright \mathrm{Hom}_H(G,X)$ to an action $G \curvearrowright X^{[G:H]}$ by isometries. Finally, notice that this action does not fix a point if $H \curvearrowright X$ is fixed-point free. $\square$
Remark: In the statement of the proposition, it is possible to replace "fixed-point freely" with "properly", but not with "geometrically" (see for instance the Coxeter group I mention here).
Now the theorem can be proved. Suppose that $H$ contains a codimension subgroup. Then $H$ acts fixed-point freely on a connected cube $X$ with a single orbit of hyperplanes, and we deduce from the construction given in the previous proposition that $G$ acts fixed-point freely on $X^{[G:H]}$, which is also a connected cube. It follows from the definition of this action that $X^{[G:H]}$ contains at most $[G:H]$ orbits of hyperplanes. Choose carefully one of them, say $\mathcal{W}$, so that the action of $G$ on the new CAT(0) cube complex $Y$, obtained by cubulating the wallspace $\left( X^{[G:H]}, \mathcal{W} \right)$, is also fixed-point free. The conclusion follows from the observation that $Y$ is again a connected cube, and that by construction it contains a single $G$-orbit of hyperplanes.
