Yes, that's exactly right. A sentence in English can easily contain one, two or even more clauses in the passive voice. For example:
No sooner had the jewelry been stolen than he was arrested.
had the jewelry been stolen is in the passive voice. That's one. And he was arrested is also in the passive voice. That's two of them now. There is really nothing stopping you from adding in as many additional clauses in the passive voice as your heart pleases.
Your example sentences can actually be rewritten as follows:
No sooner had the work been started than the door was closed.
No sooner had the work been started than he closed the door.
Because the expression you're trying to use here more often than not follows this pattern: no sooner had something happened than something else happened. I most definitely have seen that phrasing more often than the one you're using. But as far as I can tell from a simple Google search, they're both absolutely fine.