Is kinetic energy conserved separately in different directions? Or does conservation only happen for total kinetic energy? If you have a collision, the momentum is conserved separately in each single direction. Now would this same idea apply to kinetic energy? How does the nature of conservation change due to it being a scalar?
 A: Another way to look at it is that momentum is mass times velocity, and velocity is a vector: it has components in the x,y, and z directions.  Kinetic energy is mass/2 times velocity squared.  Velocity squared is the square of the length of the velocity vector, which is the same no matter what direction it is pointing; so kinetic energy does not have a direction associated with it.  
Note that this is only true in a reference system that is fixed with respect to the center of mass of the system being measured. 
If two observers are moving with different velocities past the system being observed, they will see different kinetic energies and momenta for the system.  However, as long as their velocities are constant, neither observer will see the kinetic energy of the system changing.  Also, each observer can easily calculate the kinetic energy and momentum of the system that would be observed in the system's own fixed center-of-mass system, and the observers will agree on those values.
A: Kinetic energy is not necessarily conserved, whereas total energy and momentum are.
In an elastic collision, for instance, kinetic energy is conserved as the initial kinetic energy of the particles involved is equal to the final kinetic energy of the particles involved. In this case, kinetic energy would be conserved.
On the other hand, in an inelastic collision, some of the initial kinetic energy is converted into a different form of energy, for example thermal. This means that, although the total amount of energy is still conserved, kinetic energy is not.
Concerning your points...

If you have a collision, the momentum is conserved separately in each single direction. 

Yes. This is true since momentum, a vector quantity, is always conserved. Being a vector, this means that you can pick a direction and resolve all momentum into its component in this direction and they will all "add up".

Now would this same idea apply to kinetic energy?

No. Since kinetic energy is a scalar, it is simply a number associated with an object undergoing motion. It has no directionand therefore cannot be resolved in a particular direction.

How does the nature of conservation change due to it being a scalar?

By "nature of conservation" I take it you simply mean "is it conserved or not". Kinetic energy is not always conserved just because it is a scalar. As I hope I explained properly at the start of this answer, total energy, a scalar, is always conserved. There are other scalars that are conserved, such as charge.
