Considering this is world building site instead of physics/astronomy (as pointed by Rob Watts), I would try to give an explanation that is best to my knowledge (but might not be very precise or accurate, as I am not a professional astronomer).
The magnetic field of an astronomical body is linked to the presence of electrically conductive fluid, so Hg would do the trick assuming it supports suitable convection cells.
Now there's one thing that needs to be pointed out immediately. We do not know what form mercury would take under extreme pressure. To understand what I mean, you might want to read about ice phases of common hydrogen oxide (water). Under extreme pressures, ice starts to take the form of crystals (very unexpected thing for ice!). Similarly, we don't know what form mercury would take in the core of a planet.
Depending on the size of core, the planet might get distorted out of shape if it is spinning above a threshold angular velocity as the crust would give in and crack due to the enormous pressure of circulating mercury core (atomic mass 200 versus iron that has atomic mass 56). I think it would be a horrible scene to behold, even looking at it from million miles away!
As for life, well, I don't know why would life forms be any different from what they are on Earth except that the magnetosphere plays an important role in the formation of organic molecules.