If I say: "The door opened", does it suggest that the door could have been opened both by itself or by someone?

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Literally speaking, it sounds like the door seemed to have opened by itself but it depends on the context. It doesn't completely preclude being opened by someone, but choosing not to imply any agent sounds like it could've opened on its own.

Other times it may just be because you want to emphasise the action of the door opening over who opened it.

Not so literally, this kind of sentence might also be used somewhat poetically just to describe a setting.

Here are some example sentences:

"I had to try the lock three times before the door finally opened."

"The door opened and five large men entered."

"While exploring the old house, the girl discovered a side door that opened to a beautiful courtyard."

You might find this page of example sentences from various sources useful: http://www.linguee.com/english-spanish/search?q...