From what I saw of Selvig's research and the thing itself? I think it's fully capable, but I think we'd know about it. The Tesseract is loud and kind of unpredictable in a way. I'm not sure it's responsible for this, but it's a line of thought we should consider.
Something like it, definitely. We know wormholes exist, Thor proved that one. We know the Tesseract can create a relatively manageable door between two places, possibly even two universes, if we assume the multiple universe theory exists. If that exists, I see no issue why some form of time travel also wouldn't. I have a theory that this is something from the future rather than our present day. Lots of little niggling things, like how perfectly sized the ship and the pods are - they at least knew about humans and have figured out a way to kidnap us seamlessly from our world.