Suppose that $X$ and $Z$ do not intersect transversally in $Y$. May $X\cap Z$ still be a manifold? If so, must its codimension still be $\operatorname{codim}X+\operatorname{codim}Z$? (Can it be?)
Is my solution correct?
An example I have in mind is $Y=\mathbb R^2$, $X=x$-axis, $Z=\{(x,y):y=x^2\}$. Then $X\cap Z=\{(0,0)\}$, $X$ is not transverse to $Z$, but a one-point set is a manifold. Its codimension is $2$, which equals $\operatorname{codim} X+\operatorname{codim} Z$. So the answer to the first and third question is yes.
The answer to the second question is no: Consider two orthogonal axes in $\mathbb R^3$. Each of them has codimension $2$, but their intersection has codimension $3$.