You did not explicitly ask the following, but it is related to your second question, and it comes up in practice quite often. So I would like to state this at any rate. Let $X$ be a locally Noetherian scheme. Let $D\subset X$ be a closed subscheme such that there exists a complex $E^\bullet$ of (finite rank) locally free $\mathcal{O}_X$-modules concentrated in degrees $[-c,0]$, and there exists a chain homomorphism, $$\phi:E^\bullet \to \mathcal{O}_D[0],$$ such that the induced map $$h^0(\phi):h^0(E^\bullet) \to \mathcal{O}_D$$ is an isomorphism. Moreover, assume that the restriction of $E^\bullet$ to $X\setminus D$ is acyclic. Then every irreducible component of $D$ has "codimension $\leq c$ in $X$".

Precisely, let $\eta$ be any generic point of $D$. Form the Noetherian local ring $R=\mathcal{O}_{X,\eta}$. Then the stalk $E^\bullet_\eta$ satisfies the hypotheses of the New Intersection Theorem. Therefore, the New Intersection Theorem says that the Krull dimension of $R$ is $\leq c$.

How does this come up? Here is a typical application. Let $Y$ be a regular locally Noetherian scheme, and let $C\subset Y$ be a closed subscheme that is Cohen-Macaulay and everywhere has codimension $c$. Then (at least locally), there exists a locally free resolution $$\psi:F^\bullet \xrightarrow{\text{qism}} \mathcal{O}_C[0],$$ with $F^\bullet$ concentrated in degrees $[-c,0]$. Now let $f:X\to Y$ be a morphism, let $D$ be $X\times_Y C$, let $E^\bullet$ be $f^*F^\bullet$, and let $\phi$ be $f^*\psi$. Then $E^\bullet$ and $\phi$ satisfy the hypotheses from above. Therefore every irreducible component of $D$ has codimension $\leq c$ in $X$. Note that this setup is more general than local complete intersection morphisms, the class of morphisms that arises most often in intersection theory.