I always think of 扉 as sliding doors (as also usual in trains and buses), and of ドア as a door with hinges (or some other more modern/Western construction, like elevator doors).

Except people rarely write sono with kanji

そのドアを開けて下さい

I'm pretty sure I've seen Duo use その as 'the' rather than 'that' earlier in the course, so I think you're right. I'm not an expert though.

Dropping その would make it "please open the door".

その means "that", この means "this", and あの means "that one over there". Basically, it's used to tell you where the object is in relative position to the speaker.

A lot of languages (like nahuatl, indonesian and japanese) do not have articles for "the", instead, they use the demonstrative "that", when strictly necessary to emphasize you are talking about a specific object (otherwise, they just write without it). When you are translating from these languages, "the" and "that" can have basically the same meaning, so that is something to keep in mind when learning japanese.

guess they really like to borrow stuff

i mean they got their own word apparently they just don't want to use it now a days

Please fix this. Look at the duolingo Japanese-English course. Here sono is always translated as "the".