The first is a conservation of momentum problem. Before the person throws the shoe, what is the momentum? 0 kgm/s, because nothing is moving. Now, we know the shoe is thrown and we know how fast it is being thrown. The net momentum of the system (person + shoe) before the throw has to equal the momentum of the system after the throw (because there are no net external forces acting on the system.) So:
where is the mass of the shoe, etc.
(The negative sign merely indicates that the person is moving in the opposite direction as the shoe.)
The second part of the problem is easy. It's a frictionless surface, so if the person starts moving with a velocity they will continue with that velocity. So the person is moving with a speed over a distance of 5.00 m. How long does it take?