You only need to solve the first equation, W is the work done by friction, and the system would gain kinetic energy but lose gravitational potential energy.
W = ΔEk + ΔEp
W is work due to friction.
ΔEk is the change in kinetic energy.
ΔEp is the change in gravitational potential energy.
In this case, the hat gains kinetic energy but loses potential energy, so ΔEk is positive, and ΔEp is negative.
The force the ground exerts on the person is going to last 41 cm, the initial velocity would be 0, the final velocity would be the answer to part a, from there you can solve for acceleration.
0.9m is not maximum height, so u^2 is not 17.64, but you don't need to find max height anyways.
Start with the displacement equations for both components, then put those equations together to find time.