The velocity you calculated was the max velocity for the SHM. From that and the angular frequency of the oscillator, you should be able to find the amplitude and the phase angle.
You're correct that momentum is conserved, but not energy during the collision, but after they collide, the energy of the spring plus the kinetic energy of the stuck-together masses is conserved.
Calculate the velocity of the stuck-together masses from conservation of momentum and go from there.
You got the formula correct, but did the math wrong. It should be F=4687.5N. If you draw a free-body diagram of the vehicle, you will see that the only external forces acting on the car are friction, gravity and the normal force of the road against the tires. Since gravity and the normal force...
Since you "guessed" the answer for a correctly, you can substitute that value into eq. 3 and with a little alfebra get T = mg - ma. Using that, you get T = 4.524 N. Plugging I=.5*MR^2 into eq. 4 should give you the same result.
Once the additional 7.3 kg are added and you find that the gravitational force is enough to overcome the static frictional force, you have to use the coefficient of kinetic friction to calculate that force. That should correct the problem.