I think you were confused by the misleading infomration about the spring/balance being 149cm long. The solution to the problem only requires the mass and the frequency. No "x" involved.
You are on the right track. Use the average velocity to get the time required to stop.
There is also a kinematic formula that can give you the answer directly.
Part of the important distinction is that a torque is a force applied perpendicular to the direction of movement you are referring to. Your example has the force applied parrallell to the direction of movement. The perpendicular distance is the distance to an axis of rotation which can be any...
The problem is statically indeterminate. The six equations are the six equilibrium equations: 3 force directions (in x, y, z) and 3 moment directions (about x,y,z).
If, as you suggest, we assume that the reactions are pinned rollers then the number of equilibrium directions reduces to three...
Since the applied compressive stress is 2.5ksi you need to calculate the probability that the compressive strength will be below this value. From MS Excel I estimate this should be 14.2%.
Yes, this is one of the Lagrange points. Any two body system (e.g. the earth-moon system) has five such points where the object would be seen as stationary relative to the other objects.