Hi- is there a single answer for these three questions?

What is responsible for the dissapation of gravity as you travel away
from the center of earth?  Do other planets exhibit a center of
gravity as dense (compacted)as that of the earth?  Is the earth's
center of gravity infinite or quantifiable?

thanks

I see three questions.

1.  What is responsible for the dissapation of gravity as you travel away
from the center of earth?

The force of gravity varies inversely to the square of the distance
from the center of the earth (once you are above the surface).  The
same would be true for any other planet or object.

2.  Do other planets exhibit a center of gravity as dense
(compacted)as that of the earth?

I don't know what you mean by "center of gravity as dense".  If you
are referring to what someone would weigh on the surface of the earth
as compared to the surface of other planets - on some planets you
would weigh more than on the earth and on others, less.

3.  Is the earth's center of gravity infinite or quantifiable?

The earth's center of gravity is at the center of the earth.  The
earth's center of gravity is a location, not an amount or intensity. 
However, the gravitational effects of the earth propagate outward
thruout the universe (growing weaker by the relationship described in
question 1).  There would not be a cutoff point beyond which the
earth's gravity did not reach.

Gravity gets weaker with distance for the same reason that a light
source is weaker with distance, and for the same reason that things
look smaller with distance. An object has a certain quantity of
gravitational "flux", due to its mass content, the more distant you
get from it, the less of that flux there is in a volume of space.
Think of it as Faraday pictured electric fields. Think of an object, a
sphere, with lots of straight spokes extending out of its surface.
(Like rays of sunlight.)

   If you get up close, there are lots of those spokes all crowded
together. As you get farther away, the spokes necessarily get less
concentrated. There's less flux per unit volume of space.