The right concept to solve this problem is the conservation of the Moment.
By definition the conservation of the moment is given by,
Where,
Mass of the objects/persons
Initial Velocity of each one
Final velocity of each one
PART A) For the first case the speed of the Dartboard is 0, and in the collision both objects maintain the same speed therefore when replacing the data we have to
Therefore the velocity of the dartboard moving after the dart hits and sticks is 1.29m/s
PART B) For conservation of potential and kinetic energy we have to
Where
Total mass
Re-arrange for h,
PART C) We have an inelastic collision. This understanding its definition in which kinetic energy is not conserved due to the action of internal friction.