Die werk is ernstig.

Die is the definite article in Afrikaans, equivalent to the in English. It’s used because we’re referring to a specific piece of work or job. When speaking about work in general, you can omit it entirely:
Werk is ernstig.

In predicative position (after a copula like is), Afrikaans adjectives remain uninflected. That’s why you say:
Die werk is ernstig.
If you use the adjective attributively (directly before a noun), it takes an -e:
die ernstige werk

By inverting the subject and verb:
Is die werk ernstig?
(“Is the work serious?”)

Afrikaans uses a double negation with nie … nie. You’d say:
Die werk is nie ernstig nie.
(“The work isn’t serious.”)

Yes. Without die, you speak about work in general:
Werk is ernstig.
means “Work is serious.” Use die only when you mean a specific job or task.

Ernstig means “serious” in the sense of grave or important.
Erg can be an adverb meaning “very” (ek is erg moeg) or an adjective meaning “bad/harsh” (dis erg). You generally wouldn’t say Die werk is erg to mean “The work is serious”—that could be misunderstood as “the work is very bad.”