Learn Hardcore Norwegian: Hun er en lærer.

Common gender: en lærer, en bok (a book)
Neuter gender uses et: et hus (a house)
In modern Bokmål, most nouns of people or professions are common gender, so you use en. Ei is an alternative feminine article but is rarely used separately in Bokmål (more so in Nynorsk or regional dialects).

Why is lærer treated as common gender, and are there any female-specific forms?

Lærer is a common-gender noun covering both male and female teachers.

Historically there was lærerinne (female teacher), but it’s now considered old-fashioned or marked.
Today you simply say lærer for any gender.

I’ve heard Norwegians sometimes say Hun er lærer without en—is that correct?

Yes. When stating someone’s profession, it is more idiomatic in Norwegian to drop the indefinite article:

Hun er lærer = She is a teacher (profession).
Including en (as in Hun er en lærer) can imply “one teacher among several” rather than her occupation.

How would you make Hun er en lærer definite, i.e. “She is the teacher”?

You replace en with the definite article suffix on lærer and use den before it:

Is the word order in Hun er en lærer the same as in English?

Yes, both languages use Subject-Verb-Predicate Noun in this case:

How would you say “She is a teacher” in Nynorsk?

In Nynorsk you use ho for “she” and lærar (nynorsk spelling), so:

Ho er lærar (you usually omit the indefinite article for professions, just as in Bokmål).