Your questions are hard! I got help from my brother, who does mapping with satellites and computers.

He says we measure the density of the earth with a gravitometer. That´s the short answer. "GeoMan," on the internet, has another answer: He says scientists estimate the density of the earth from the mass of the earth and the volume of the earth. Density is the mass divided by the volume.

Scientists calculate the mass of the earth from an equation of Newton's that calculates the mass from the gravitational attraction of the earth to some other large object. Scientists calculate the volume of the earth from the radius, which GeoMan says is easy to find. I suppose it's very easy to measure the radius of the earth now, because we have pictures of earth from space! The volume of a sphere is equal to 4/3 times Pi times the radius cubed (4/3π R³).

But that only gives an approximate answer for the average density of the earth. As my brother says, you can also think about the different densities of different mountains! The Grand Teton is granite, so very dense, but Mount Rainier is volcanic and porous.

Here´s what my brother says about your second question, with some more info about measuring density at the end:

Small areas of the earth have been measured and mapped for thousands of years by terrestrial methods. That is to say, measurements made "on the ground". Linear and angular measurements are made. Linear measurements were made with sticks, ropes, poles or chains. Nowadays linear measurements are made with light. Angles are measured with protractors, compasses, transits and theodolites. Transits and theodolites are simply large protractors with telescopes attached.

Large areas of the earth, including the entire earth, are measured and mapped by celestial methods. For thousands of years this was done entirely by observing the location of natural heavenly bodies (the sun, stars and planets). More recently a constellation of man-made satellites known as the Global Navigation Satellite System (GNSS) is used. As the name suggests they are often thought of as primarily for navigation, but they have become the standard method for mapping large and small areas of the earth. And they are accurate enough to measure the movement of the continental plates!

Navigation and mapping are closely related and use many of the same techniques and tools. Geodesy is the science of mapping the earth. In addition to mapping the size and shape of the earth, geodesy studies the density of the earth. Density effects the gravitational pull which is important to everything from the direction that water flows to the trajectory of a rocket