Learn Spanish (Latin America): Él duda que ella mienta.

Because the first part of the sentence expresses doubt (Él duda que...). In Spanish, when a sentence has a verb of doubt, disbelief, or denial in the main clause, the verb in the second clause (after que) must be in the subjunctive mood to show that the action is not being stated as an objective fact.

Where does the "ie" in mienta come from if the infinitive is mentir?

Mentir is a stem-changing verb where the "e" changes to "ie". In the present subjunctive, verbs follow the exact same stem changes as they do in the present indicative for most forms. Since the normal present tense for "she lies" is ella miente, the subjunctive keeps that "ie" stem change to become ella mienta.

The English translation says "she is lying". Do we need to use a continuous tense in Spanish, like esté mintiendo?

You don't have to! The simple present subjunctive (mienta) can translate to either the simple present ("she lies") or the present continuous ("she is lying") in English, depending on what makes the most sense in context. You could say Él duda que ella esté mintiendo, but using the simpler form mienta is extremely common and natural for both meanings.

Is it completely necessary to include the pronouns él and ella in this sentence?