Dear Long,
to give a shamelessly trivial answer to your question 2, I would say that the point is usually that the failure of these properties is closed, often "obviously". I would also add one more auxiliary property, that acts as a meta property for some of these:
A coherent sheaf being locally free is an open property. This follows from Nakayama's lemma.
So, here is your list. The properties on the left fail along the loci on the right. [Caveat: I did not include conditions that are sometimes necessary, but I figured that this is a philosophical question and so the answer does not have to be stated in the most precise way.]
regular/smooth -------------- zero set of the Jacobian ideal, also the locus where $\Omega_X$ is not locally free
Cohen-Macaulay, $S_n$ ----- support of appropriate Ext sheaves
Gorenstein ------------------ {not CM} $\bigcup$ {CM but $\omega_X$ is not a line bundle}
$\mathbb Q$-Gorenstein --------------- $\bigcap_m$ {$\omega_X^{[m]}$ is not a line bundle}
rational singularity --------- (in char $0$) $\bigcup_{i=1}^{\dim X}{\rm supp\\,}R^i\phi_*\mathcal O_{\widetilde X}$$\bigcup_{i=1}^{\dim X}{\rm supp\,}R^i\phi_*\mathcal O_{\widetilde X}$ where $\phi:\widetilde X\to X$ is a resolution.
klt singularity ---------------- zero set of the multiplier ideal.
Du Bois singularity --------- $\bigcup_{i\neq 0} {\rm supp\\,} h^i(\underline\Omega_X^0)\bigcup \\,{\rm supp\\,}{\rm coker\\,}[\mathcal O_X\to h^0(\underline\Omega_X^0)]$$\bigcup_{i\neq 0} {\rm supp\,} h^i(\underline\Omega_X^0)\bigcup \,{\rm supp\,}{\rm coker\,}[\mathcal O_X\to h^0(\underline\Omega_X^0)]$
(semi-)normality ------------ ${\rm supp\\,}{\rm coker\\,}[\mathcal O_X\to \pi_*\mathcal O_{Y}]$${\rm supp\,}{\rm coker\,}[\mathcal O_X\to \pi_*\mathcal O_{Y}]$, where $\pi:Y\to X$ is the (semi-)normalization.
etcetera...