"decreasing the distance of the space shuttle from Earth"
F = Gm(1)m(2)/R²
where R is the distance between the 2 objects, as it decreases, the force increases.
Yes, if the mass starts at rest, <u>the change in speed will be equal the final speed</u>, because:
Δv = Vf - Vo
How Vo (Initial velocity) is equal zero, we simplificate:
Δv = Vf
Then, the change of the speed, if the mass starts at rest, will be equal to final velocity.
Greetings.
No force contributes to density, the density is a physical quantity that is defined as being

the raport between the mass of the object and its volume. However if you want to measure the density of an object you might want to determine its gravity force (weight)

from which knowing the gravitational acceleratin you can find its mass
where

is given in Newtons and

is given in