A horizontal line on a speed/time graph means a constant speed.
A force of 43.8 N is required to stretch the spring a distance of 15.5 cm = 0.155 m, so the spring constant <em>k</em> is
43.8 N = <em>k</em> (0.155 m) ==> <em>k</em> = (43.8 N) / (0.155 m) ≈ 283 N/m
The total work done on the spring to stretch it to 15.5 cm from equilibrium is
1/2 (283 N/m) (0.155 m)² ≈ 3.39 J
The total work needed to stretch the spring to 15.5 cm + 10.4 cm = 25.9 cm = 0.259 m from equilibrium would be
1/2 (283 N/m) (0.259 m)² ≈ 9.48 J
Then the additional work needed to stretch the spring 10.4 cm further is the difference, about 6.08 J.
The centripetal force needed to keep earth in orbit is gravity.
That's true.
Netwon's second law states that the resultant of the forces F acting on a body is equal to the product between its mass m and its acceleration a:

This means that if the net force acting on an object is different from zero (term on the left), than the acceleration of the object (term on the right) must be different from zero as well, and therefore the body is accelerating.
In particular, both F and a in the equation are vectors: this means that if the acceleration is positive, F and a have the same direction. In this problem, the acceleration is positive (because the object is speeding up), therefore the force and the acceleration have same direction.