Pushouts in the Category of Schemes When does it make sense to glue schemes together along subschemes?
In particular: is there a way to glue two schemes together along a closed point (say we're working over a field)? Can you glue two closed points of the same scheme together? 
Is it easier to glue in the category of algebraic spaces?
 A: 
is there a way to glue two schemes together along a closed point (say we're working over a field)? Is it easier to glue in the category of algebraic spaces?

For this particular pushout, the geometric intuition is quite simple: given two algebraic varieties, one of which lives in $\mathbb A^m$, another in $\mathbb A^n$, combine them in two complementary hyperplanes in $\mathbb A^{m+n}$. Algebraically, this easily generalizes to an affine scheme $\mathrm{Spec}(R_1\times R_2/\mathrm{relationship})$ and then you glue everything together.
As correctly said above, general pushouts of schemes may not be schemes themselves.
A: Since it doesn't seem to have been mentioned yet, Ferrand proves in Theorem 7.1 of "Conducteur, descent, et pincement" that you can pushout closed immersions $Y' \to X'$ along various nice affine morphisms $g: Y' \to Y$. 
For example, if $g$ is finite, and every finite set of points of $X'$ and $Y$ are contained in an open affine (e.g., they are projective varieties) then the pushout exists (Theorem 5.4 of loc.cit.). 
A: In the affine case, let $X=\text{Spec }A$, $Y= \text{Spec }B$, and $Z= \text{Spec }R$. If you have morphisms $f:Z\rightarrow X$ coming from $\phi:A\rightarrow R$ and $g:Z\rightarrow Y$ coming from $\psi: B\rightarrow R$ (because $\text{Aff}$ is anti-equivalent to $\text{CRing}$), then the pushout $X \coprod_{Z} Y$, gluing $X$ and $Y$ along $Z$ is given by $\text{Spec }D$, where
$D=A\times_{R} B:=\{(a,b) \in A\times B \mid \phi (a)= \psi (b)\}$.
A: Given schemes $X,Y$ and $Z$ such that $Z$ is a closed subscheme of both $X$ and $Y$ the pushout exists in the category of schemes. So in particular one can glue schemes along a closed point. A reference for this (carried out via the category of locally ringed spaces) is given in  this paper of Schwede (Corollary 3.9).
In general though the pushout in the category of locally ringed spaces need not be a scheme even if one pushes out along a subscheme - see for instance Example 3.3 in Schwede's paper.
A: Consider a commutative local ring R, say a valuation domain, with maximal ideal M. Consider the fiber product $R \times_M R$ (I wrote M instead of R/M), coming from the pullback in commutative rings $R\rightarrow R/M$. Then the corresponding prime spectra of this fibered product (in rings) is actually a form of gluing of the same same (affine) scheme Spec R along the closed point M. So this is the case where this happens. 
So I think you can do such things for affine Schemes. For affine schemes, you can at least reverse the topology (they are sometimes called inverse spectrum) and you can form a sheaf over this topology similar to the canonical structure sheaf, but the closed points becomes the generic points in this topology. I cannot recall correctly, but I think the stalks of this sheaves become integral domains (so it is some form of dual to the affine schemes, local becomes integral and so on)
