How Centre of Gravity (CG) is adjusted?
The main issue is what is referred to as the Weight and Balance of the aircraft. When an aircraft is in flight, its weight and balance needs to be within a controllable range in order to ensure the plane can both recover from a stall condition and rotate on takeoff. This is visualized by the weight and balance "envelope:



As you can see, there are limits where the aircraft is not fit for flight. We must stay inside the drawn lines in the center.

To answer your detailed questions:


On airlines, a "Magic" number was used to approximate all weight for passengers and baggage. This caused at least one  notable accident: Air Midwest Flight 5481. As a result of this accident, policies were changed to get more accurate baggage and passenger weight.

In GA flight, it is not uncommon for pilots to ask for passenger weights. Usually, as a rule, we add a bit to that in case people are embarrassed (10%). It is very common for pilots to assign seating on smaller aircraft with heavier people sitting in a location where the distance to the datum is smallest (near the front).
Weight and balance can be moved out of limits by fuel burn. In fact, a few questions on the private pilot written exam deal with this exact scenario. Most planes have tanks near the CG, so they don't experience problems with fuel burn. In planes with an aft belly tank, the CG can move forward significantly and out of limits. This would cause the aircraft to be unable to pitch up as desired. Swept wing jets that use horizontal stabilizer tanks in the tail of the aircraft have computers and pumps to move the fuel as desired to keep their CG in the desired place.

Pilots of smaller/less traditional airplanes need to be conscious of this, and plan accordingly. Especially those with a-typical fuel tank positions. Pilots of larger commercial style aircraft are less concerned with the exact position of 1 or 2 large people or heavier bags because their aircraft has a much bigger capacity and less "sensitive" envelope. They still are affected by the physics, though, just like everyone else.