Pushout exists in category of pointed topological spaces It is not mentioned anywhere but do push out always exists in pointed spaces?  (I will omit the points below)  We have something like 
$\require{AMScd}$
\begin{CD}
    X @>f>> Y\\
    @V g V V @VV  V\\
    Z @>>> Y\vee Z/\sim
\end{CD}
I claim the push out is $Y \vee Z / \sim$. 
where $Y\vee Z$ is the pointed space, $Y \sqcup Z / *_Y \cup *_Z $. 
$\sim$ is the relation generated by $i_Yf(x)\sim i_Zg(x)$ for $x \in X$. $i_X,i_Z$ being inclusions. 
Proof: A cocone of the push out diagarm, with maps $g':Z \rightarrow A, f':Y \rightarrow A$ induces a map $Y \vee Z \rightarrow A$. But the conditions that it is a cocone induces a map $Y \vee Z /\sim \rightarrow A$. 
 A: For any category $\mathcal C$ and any object $c$, a diagram $D: J \to c\backslash\mathcal C$ in the coslice category induces a diagram $\tilde D : \tilde J \to \mathcal C$ where $\tilde J$ is the category $J$ with a formal initial object $\varnothing$, such that 
$$ \tilde D(\varnothing \to j) = D(j) \qquad \tilde D (k \overset f \to k') = D(f) \qquad \forall j, f:k\to k'\in J$$
(This is just formally saying that if you draw a diagram in $c\backslash \mathcal C$, then you can look at the "same" diagram and view it in $\mathcal C$.) 
Then colimits of $D$ coincide with those canonical maps $c\to K$ where $K$ is a colimit of $\tilde D$. It is more or less tautological and reduces to the definition of morphism in $c\backslash \mathcal C$.

Take now $\mathcal C = \mathsf{Top}$ and $c = \{\ast\}$. It tells you that the pushout of $(Y,y) \overset f\leftarrow (X,x) \overset g\to (Z,z)$ exists in pointed topological spaces and is $(Y\sqcup_X Z, \ast)$ where $\ast$ is $z=y$ in that space. So your construction seems OK, but there is no need to take first the quotient by $y\sim z$ as it will be done by the latter $f(x)\sim g(x)$ (remember $f,g$ are pointed maps so they must take the distinguish point to the distinguish point).
A: Most categories of "all" mathematical objects of a given type have all (small) limits and colimits. Examples are categories of algebraic obects, in which limits are constructed as in sets, or a more general underlying category, and colimits are constructed out of the free algebras and categories of topological objects, as you have here. Unpointed topological spaces have all limits and colimits constructed as in sets, with an appropriately maximal or minimal topology. Pointed topological spaces then inherit all limits and colimits: limits and connected colimits, such as pushouts, are constructed as in spaces, while for non-connected colimits we also have to identify the basepoints, as in the wedge product. This can be seen as a special case of the story about algebraic objects mentioned above: a pointed object is an algebra for the very boring theory that says "there's a chosen element and no operations." 
