The hint on, "открываю," states that it means, "Do I open?" Does that mean the word alone can be used as a question?
No. The hint shows that "Я открываю", if used as a question, means "Do I open".
I think it's because the door is a inanimate masculin thing (or neutral noun, I don't know). For an animate masculin thing (lion or other) you must decline to accusative masculine. For feminine noun its different.