Learn English – What does “Do you even North America?” mean

Even in this context is an adverb that is

used to emphasize something surprising or extreme : they have never even heard of the U.S. | they wore fur hats, even in summer. [NOAD]

It is merely a kind of intensifier, usually used to express the outrageousness of a thing. Now let's get down to cases:

Are you even listening?

The speaker doubts you are listening and is rebuking you for your inattention.

What time does the shop even close?

This one's a little harder, but probably someone has suggested the speaker go to this shop and the speaker doubts that it's open, or else someone has suggested that shop may be closed but the speaker is emphasizing the fact that the shop in fact stays open quite late. It would depend on context.

Just to add to the comment by Joe, as a young American, I hear "What are you into?" as

• What are your hobbies?

• What vices do you indulge in? (this may have sexual innuendo depending upon context)

• What are you doing right now in your profession/career/studies. ("I'm learning philosophy.", "Oh, what are you into?", "Well, ethics, mainly." ...)

A related colloquialism is "what are you up to?" which means "what have you been doing recently?".