When speaking of gravity, you should refer to volume, not area, as all matter is 3-dimensional (or, rather, *at least* 3-dimensional). To all intents and purposes, gravity can be taken as acting from the centre of an object. Therefore, as the diameter decreases the force at the surface will increase.

For instance, if an object has an original radius of 4,000m and, somehow, gets shrunk to a radius of 2,000m, the force exerted 4,000m from the centre will be the same as it was when the surface was there. At the new surface, 2,000m from the centre, the force will be 4 times that at the old surface (double the distance = half the strength).

In the case of a star, as it contracts, more of its mass gets closer to the centre and, hence, feels a greater gravitational force. It will therefore collapse further causing more mass to get closer to the centre, and so on.

As far as I am aware, black holes do not have magnetic fields. So, somewhere between being a star & being a black hole, the magnetic field disappears. I'm not sure at what point that happens.