How can kinetic energy not be conserved if momentum is conserved in a system? If there is no net force on a system, energy is conserved. 
Then, how can there be work at all since work is the integral of dot product of force and $dr$ ? If force is $0$, wouldn't work automatically be $0$ , as well? I realize that it makes sense if I calculate $KE$ with $(mv^2)/2$, but with the work formula, it doesn't really make sense for me...
 A: If the external force on a system is equal to zero, then its total momentum is constant.  However, one could still have internal forces (such as the forces between particles during a collision) which could change the total kinetic energy of the system.
A: Two equal size pieces of wet clay fly toward each other at equal speed. The total momentum is $0$, but there is kinetic energy. 
The pieces collide and stick together. The speed of the bigger lump is $0$. The total momentum is still $0$. But kinetic energy has been used to deform the clay. The clay is a little warmer after the collision because the kinetic energy changed to another kind of energy. 
A: There is a good summary from this book :
D. Griffiths, Introduction to Elementary Particles, 1987, John Wiley & Sons, section 3.4, pp. 91-93.
Classical collisions:
1. mass & momentum are conserved
2. kinetic energy (Ek) may or may not be conserved
Types of classical collisions:
1. sticky: Ek decreases
2. explosive: Ek increases
3. elastic: Ek conserved
Relativistic collisions:
1. energy-momentum 4-vector is conserved
2. Ek may or may not be conserved
Types of relativistic collisions:
1. sticky: Ek decreases, rest energy & mass increase
2. explosive: Ek increases, rest energy & mass decrease
3. elastic: Ek, rest energy & mass are conserved 
A: Sum of change in kinetic energy and change in potential energy is called work done.If there is no change in their energies,then there is no work have been done.A momentum has to be imparted to change its state of rest or uniform motion.Work is related to energies by work energy theorem.
In your above case , work is done in giving as kinetic energy.SO IT IS FINE.
