1) y= h(x)

The distance ds of a tiny piece of the cable is given by

Get dy in terms of dx and put it into that equation so that you have ds in terms of dx only. Then integrate ds between x=-10 and x=10.

2)Let the rate at which mass leaves the bucket be k where k is in kilograms per meter

Since the mass leaves at a constant rate the mass as a function of distance is

M(s)= 20-ks

You can find k with the information you are given.

You know that work= force*distance, then

dW= Fds

The force is equal to M(s)a= 9.8M(s)

Integrate

dW= 9.8 M(s)ds

The velocity isn't needed, they probably expected you to get the rate which mass leaves the bucket in kilograms per second and then get that in terms of distance using the velocity but this step isn't required.