I know some ways to change how quickly an object can we affected [traslation] and such, but how can I change the overall weight so it gets affected differently by gravity?
Realtime mass changes can be accomplished with the "Edit Object" actuator (my fav) set to Dynamics>Set Mass.
However, this still doesn't make the object fall faster(Galileo found this ~a long time ago :D)! think, what is gravity, it's a force, right? So applying more force means an object falls faster. If an object is dynamic or rigid body, you get alot more buttons to play with under "Motion". Try adding a force on the -Z axis, remember, 9.80 = 1G. Make sure to uncheck the "L" for local; that way you fall down, not toward your feet. Although they are sometimes the same place, if the player gets flipped, you have problems.
Heavier objects don't fall faster, so you shouldn't be changing the mass. instead decrease the damping factor ( which is like air resistance when falling ) and the object will go faster. also if there is a way to change the gravitational force effect on this object it would work if you increase it.