Wow... Just realised how big that question is. Let me summarise for those who can't be bothered reading it.
What is the solution to this second order linear non homogeneous differential equation?
M*(d2x/dt2) + C*(dx/dt) + K*x = Mg
My answer is:
x(t) = C1(e^pt)+C2(e^qt) + Mg/K
But this is obviously wrong because it does not describe oscillations.