Translation:Please open that door.
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).
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.
As far as I'm concerned 'the' is a usual translation of 「その」. That's what I typed and got it wrong. As this remains as a beta version we'll have to cope with this kind of limitations, but it is very frustrating. Report as much as you can!
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.
i mean they got their own word apparently they just don't want to use it now a days
"Please open the door." Should be right. There is no need to translate その as that.