Is every finite field a quotient ring of ${Z}[x]$? Is every finite field  a quotient ring of ${Z}[x]$? For example, how a field with 27 elements can be written as a quotient ring of ${Z}[x]$?

Every finite field has an order which is the power of a prime.
Every finite field of order $p$ is isomorphic to integers modulo $p$. Every finite field of order $p^k$ is isomorphic to polynomials over the field with p elements; modulo an irreducible polynomial of degree $k$. There are no other fields.
So yes. Taking $\mathbb Z[x]$ through first a quotient on $p\mathbb Z$, then on $d\mathbb Z[x]$ with $p, d$ prime/irreducible will yield a field.
