= Solution
\b[All three assertions are false.] For the transversality assertion, use $f:\mathbb R\to\mathbb R$, $f(x)=x^3$, and $Z=\{0\}$. At the preimage point the derivative is zero, so $f$ is not transverse to $Z$, yet $f^{-1}(Z)=\{0\}$ is a codimension-one submanifold. Transversality is sufficient, not necessary.
For openness of <regular values>, the smooth function $f(x)=e^{-x}(2+3\sin x)$ gives a connected-domain counterexample. Every zero has $\sin x=-2/3$ and derivative $3e^{-x}\cos x\ne0$, so zero is a <regular value>. <Critical points> solve $\cos x-\sin x=2/3$. Choose any such point $x_0$; its value is nonzero, since simultaneous vanishing would force $\sin x=-2/3$ and $\cos x=0$. The <critical points> $x_0+2\pi n$ have nonzero critical values $e^{-2\pi n}f(x_0)\to0$. Thus no neighborhood of the <regular value> zero consists entirely of <regular values>. Properness would supply the missing compactness needed for the usual open-regular-value conclusion.
For the measure assertion, a constant map $f:\mathbb R\to\mathbb R$ has every point critical. Its critical-point set has positive, indeed infinite, measure. <Sard theorem> concerns the measure of the set of critical values in the target, not the measure of <critical points> in the domain.
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