As a native speaker of English, I also find their translation incredibly awkward. It should definitely be "The door was shut, but I had the key." From what I understand, we should use the present tense for the second verb here in Russian because the action hasn't been concluded? In English, however, the first verb puts the entire sentence into the past, so the second verb should also be in a past tense (though exceptions do obviously occur).
"The door was shut" means that the door was closed in the past, but the Russian sentence means that the door has shut right before the moment when an interlocutor said this phrase, so that is why the first part of the sentence (in Russian) is in the past tense (because the door is already closed) and the second one (also in Russian) is in the present (because the interlocutor currently has the key).
This is a reflexive verb in the past, denoting an action. I think it means that the door closed (by itself, in the past or just now) but I can open it now because I have a key. "The door shut" is not very idiomatic, so it should be removed. "Closed" is better as an intransitive English verb to stand for the Russian reflexive.
Seems like this comment makes the most sense to me. In other words the translation is "The door shut (itself), but I have the key."
I was looking for a reflexive example of "shut" and found "the door shut behind him," and that makes perfect sense in English. So there are idiomatic and reflexive uses of the word "shut," however not in this case. I agree "closed" would be better then.
Minor quibble: the verb is intransitive, not reflexive: https://en.wiktionary.org/wiki/shut#Verb
"The door shut" is elegant, concise, and idiomatic in some English variants, but can lead to confusion when used out of context. As demonstrated.
"It was a dark and stormy night. Heathcliffe ventured out to the porch to get the newspaper. A gust of wind shook the house. The door shut. He was trapped outside, his coffee slowly cooling beside the television." (to be continued)
The English translation on this one makes no sense. If the door WAS SHUT then it implies that it is NOW OPEN thus rendering a key useless (unless one wants to relock said door) So the sentence should have been:
The door WAS shut but I HAD the key. OR The door IS shut but I HAVE the key.
Anything else just sounds really wrong. That being said, as a native speaker, I would say LOCKED if I felt the need to then mention that I have/had a key. Otherwise the mention of a key is pointless. A SHUT door does not, in English anyway (north American), automatically imply that said door is LOCKED. If someone says to you "Go out and shut the door behind you." That does not mean that you will LOCK the door as leave but simply CLOSE it thus implying that the door will be shut but UNLOCKED.
Front doors aren't like that everywhere. Where I lived in Russia, for example, a lot of people had two doors to enter their apartment or house: a heavy, metal outer door that blocks the cold, often locks with a skeleton key (loved those!), and is always locked until you use the key, like it's spring loaded and you have to twist and hold the key to open the door, and when you release the key it locks again; and then inside of that heavy door within the same door frame is a regular inner door, both with heavy insulation around them. (It's cold in Siberia!) But in most places in North America (I won't assume that colder areas in Alaska and Canada don't have that kind of set up as well), we have a single regular door with a deadbolt you can lock and unlock. It doesn't have a spring-loaded mechanism to keep it locked at all times. So doors can be shut but not locked. And I would add that bedroom doors, bathroom doors, classroom doors, etc., aren't necessarily locked when they are shut. So it doesn't make sense to assume this sentence means the door locked when it closed. It does imply that meaning when you mention you have the key, but it's still awkward.
The sentence in the Russian language has this sense, that I stay by the open door, but now, as for example the wind closes the door and I say "Дверь закрылась". It is happening now. if the door WAS shut i say "Дверь была закрыта", if the door locked "дверь закрыта" or maybe "дверь заперта"