端 and 隅: an edge and a corner area
Compare a desk’s edge with a room’s corner, using the object’s shape rather than limiting 端 to corner points.
An end or edge
端, read はし, names an end or edge. 机の端にコップを置かないでください。 asks someone not to put a cup at the edge of the desk.
紙の端を折りました。 says that an edge of the paper was folded. It does not specifically say that all four corners were folded.
An area in a corner
隅, read すみ, names a corner area of a room or object. 部屋の隅に傘を立てました。 says that an umbrella was placed upright in the corner of the room.
箱の隅に小さな部品が残っています。 says that small parts remain in a corner of the box. This is different from a position outside the box.
Using the object’s shape
丸い机の端にも物を置けます。 says that things can also be placed at the edge of a round table. 端 is not limited to angular corners.
端から端まで線を引きました。 says that a line was drawn from one end to the other. Read から and まで for both endpoints.
Does 紙の端を切りました。 necessarily mean cutting only the four corners?
Show the answer and why
No. It says that an edge area was cut; the specific part needs more information.