References and background on universal homeomorphisms
Definition [EGA I (2d ed.) 3.8.1]. A morphism $f:V\to U$ is a universal homeomorphism if for any morphism $U'\to U$, the pullback $V\times_UU'\to U'$ is a homeomorphism.
Theorem [EGA IV 18.12.11]. A morphism is a universal homeomorphism if and only if it is surjective, integral, and radicial.
Theorem ["Topological invariance of the étale topos," SGA I Exp IX, 4.10 and SGA IV Exp. VIII, 1.1] If $f:V\to U$ is a universal homeomorphism, then the induced morphism $f:V_{\textrm{ét}}\to U_{\textrm{ét}}$ of the small étale topoi is an equivalence.
General examples. Any nilimmersion, any purely inseparable field extension (or any base change thereby), the geometric Frobenius of an $\mathbf{F}_p$-scheme [SGA V Exp. XIV=XV, § 1, No. 2, Pr. 2(a)].
Theorem. Suppose $X$ a reduced scheme with finitely many irreducible components. Denote by $X'$ its normalization. Then the natural morphism $X'\to X$ is a universal homeomorphism if and only if $X$ is geometrically unibranch.
Specific example. Suppose $k$ an algebraically closed field of characteristic $2$. Consider the subring $k[x^2,xy,y]\subset k[x,y]$. The induced morphism
$$\mathrm{Spec}k[x,y]\to\mathrm{Spec}k[x^2,xy,y]$$
is a universal homeomorphism.
Question
Do pushouts along universal homeomorphisms exist in the category of schemes?
In more detail. Suppose $f:V\to U$ a universal homeomorphism, and suppose $p:V\to W$ a morphism. Everything here is a scheme; I can assume $W$ quasicompact and quasiseparated, but I have no control over the map $V\to W$. Now of course I can construct the pushout $P$ of $V\to U$ along $V\to W$ as a locally ringed space with no trouble (just take the underlying space of $W$ along with the fiber product $O_W \times_{p_{\star}O_V}p_{\star}O_U$), but I can't show that $P$ is a scheme. Is it?
Thoughts
Of course the key point here is that $f$ is a universal homeomorphism, not just some run-of-the-mill morphism. So one can try to treat the cases where $f$ is schematically dominant or a nilimmersion separately.
Update
If $f$ is a nilimmersion, then I now see how to prove this completely. I still have no idea how to proceed the schematically dominant case.
[EDIT: I removed the additional question.]
Best Answer
The answer to your question is yes in positive characteristic. Check Kollár's paper "Quotient Spaces Modulo Algebraic Groups" (Ann. of Math. 1997), Lem 8.4. The reason is that universal homeomorphisms behave as nil-immersions in positive characteristic (due to Frobenius). I think the answer is no in characteristic zero.
A general comment is that without assuming that one of the two morphisms in a push-out is a monomorphism the "correct" push-out should be a stack. This partly explains why most results on push-outs are when one of the maps is a monomorphism (e.g. push-outs of a finite map and a closed immersion, push-outs of an etale morphism and an open immersion).