Near the ends of a bar magnet, the direction of a compass needle will align itself with the magnetic field lines and point towards one of the magnet's poles. If you place the compass near the north pole of the magnet, the compass needle will point north, and if you place it near the south pole, the needle will point south.
In the region near the center of the magnet, the compass needle will not strongly align with either pole. Instead, it will be less affected by the magnetic field, and the needle may not point in a consistent direction.
Regarding field lines that run off the paper, typically when representing magnetic field lines using a diagram, lines running off the paper indicate field lines that are directed away from you, perpendicular to the surface of the paper. Conversely, field lines that run into the paper indicate magnetic field lines directed toward you. So, field lines running off the paper from a magnet's north pole would indicate the magnetic field lines spreading out into space, and those running into the paper from the south pole would indicate the field lines converging into the magnet. These field lines represent the path a hypothetical north pole would take when moving in the magnetic field.