IN the corner if it is where you are putting something that is closed - like in the corner of the page. AT the corner is if you are talking about a location - like the corner of a street.
AT - if you are at a place, you might not be in it. You might be at the University to see a show. You will be IN University if you are studying as a part of your University studies.
In this instance, you would use <em>in </em>university. <em>At </em>is only used before specific terms such as "I am studying <em>at </em>Harvard" or "I am studying <em>at </em>Williams College."
For the second question, context must be applied. If you are talking about the corner of a room, <em>in </em>should be used. However, if you are talking about the corner of a sidewalk or street, <em>on </em>or <em>at </em>would be correct.