Thanks for the confirmation and correction, kib!

By the way, with regard to normal English usage of participles, I would say that a present active participle construction such as "I saw a car going on a red light" would be completely normal even in the most colloquial speech (i.e., not at all "bookish"). Past passive participles ("The boy learned a dirty word from graffiti written on a wall.") are also very common in speech. And now that I think about it, present passive participles are also sometimes used in colloquial speech ("The police were looking for a green van being driven by two Middle Eastern men.").

However, past active participles are rare or non-existent in English, even in the most formal writing -- one must, instead, use a relative construction with "that" or "which." (I just Googled on "past active participle," and ALL of the hits were discussing non-English languages.) Theoretically, the past active participle would look like "The soldier brought a message from the regiment having taken the enemy position" -- but that simply isn't used in English, either in writing or speech. Instead we can only say "He brought a message from the regiment that had taken the enemy position."

As to the English equivalents of the деепричастие, where a participial clause functions adverbially: a construction like "Having left the house, I saw..." is possible in speech, but sounds slightly bookish -- although one might expect to hear it in speech from a witness giving testimony in a courtroom, for example.

But in informal speech, "After I left the house, I saw..." would sound much better.