For number 6 conserve momentum mV = (m+M)u and then conserve energy (even though the collision is inelastic you know where the extra energy went; into compressing the spring) 1/2 mV2 = 1/2(m+M) u2 + 1/2 k d2.
7: A piece of clay falling into two chunks is an inelastic process so you can only conserve momentum to solve this problem. It is a two-dimensional decay, so you need to write the equations for the x and y directions. This gives 2 equations for 2 unknowns (M and theta). Use m1i = M+1kg , m1f = 1 kg, m2f = M, eliminate M. The resulting equation looks like A sin(theta) = cos(theta) - B, where A and B are known numbers. Square this and use 1 = sin2 + cos2 to get a quadratic equation for cos(theta):
-(1+A2)cos2 + 2 B cos + A2 - B2 =0
Finally, solve this for cos(theta).