They are in orbit around their common center of mass. Gravity and centrifugal force are in balance.
weight = mass x gravity
Since gravity is produced by mass, the center of mass is also the center of gravity. The only difference between these two concepts is that mass is a more basic quantity, so the center of mass would also be the center of inertia, as well as the center of gravity. In practice, these terms can be used interchangeably.
Mass is uniformly distributed about its center of mass.
i have no clue
They orbit about their mutual center of gravity, which is in between the two because they have similar mass. Other planets have the center of gravity inside them.
The relationship is:weight = mass x gravity
The strength of gravity between 2 bodies depends on your mass and the planet's mass, and the distance between the center of your mass and the center of the planet's mass.