"Stay with him until he does the work."

Translation: הישארי איתו עד שהוא יעשה את העבודה.

[TeribleT]

Hisha'ari* ito ad she'hu ya'ase et ha'avoda

*(to a woman)

[rzil]

"הישאר איתו עד שיעשה את העבודה" should be correct as well, as (mostly) in hebrew the verb also contains the subject

[mgrdAT]

No. The pronouns אני אתה את אנחנו אתם אתן (first and second person) are not needed in past and future tense, as far as I know (but duolingo often insist on using/requiring אני in first person future tense). The third person pronouns הוא היא הם הן are usually required in all tenses.