How is momentum conserved when is is only dependent on mass and velocity, and so many other factors come into play? I've been trying to get a good grip on the difference between conservation laws. Momentum is particularly tricky, I don't understand how quantities like $m\mathbf v$ can be conserved when other things like deformation and heat come into play. Googling it seems to suggest that the answer lies in energy being a scalar and momentum being a vector, but then this situation still confuses me: 
Two cars with the same mass and different velocities, say $-5\mathbf i$ and $10\mathbf i$, collide and no external forces act.
All their kinetic energy goes into sound, heat, and deformation. So final velocity is reduced to zero for both of them. 
What happened to the momentum? Their vectors were unequal so it can't be zero, but the final velocities are zero.
Thanks for any help.
 A: The momentum of the composite of two cars is the vector sum of the momentum of car 1 plus that of car 2.
That composite momentum is the same before and after the collision.
This assumes they are colliding above the ground, or they are on a frictionless surface.
Example: two 1kg sticky bean-bags are moving together. One is traveling to the right at 3m/s.
The other is traveling to the left at 1m/s.
The composite center of mass is 2kg traveling to the right at 2m/s.
After the collision, you have a 2kg mass moving to the right at 2m/s.
On the other hand, if they are elastic they will bounce apart and be two separate masses again.
But you can rely on it that their composite center of mass is just as heavy and continues in the same direction, with the same speed, as it did before the collision.
If they are on the earth's surface and there is friction, so they stop, then you also have to consider the momentum of the earth in the composite as well.
Since you probably don't want to do that, that's why people make idealizing assumptions.
A: Your assumption (intuition?) that the resulting lump would be not or little moving is simply wrong. The resulting lump would move with 2.5i, as we can easily calculate:
-5i * m + 10i * m = 5i*m  -> that is the total momentum before and after the impact.
As the total mass of the lump after the impact is 2m, it moves with 5i*m / 2m = 2.5i.
If this is hard to imagine, then consider the case that car 1 is going 80 miles per hour, and car 2 is just rolling slowly, with only inches per second. Do you really think after the impact the result would sit still? The lump would move with just short of 40 mph in the same direction (after a nice loud bang).
A: I'd imagine the momentum went in to the air molecules too, no? That and the molecules of the asphalt. Perhaps the friction is like a huge number of micro-collisions occurring, so the momentum is gradually transferred to the asphalt molecules while the cars together slow down.
A: Yes! I think momentum can be zero in case you consider air as suggested by Mr. SSD. You can imagine the scene, as in initial case the both cars are moving and wind isn't blowing. Then two cars collide and  lose their KE and momentum to air. Then a soothing warm (because of heat of collision) wind blows(to conseve the momentum) with the two cars lying at rest.
In that case you have to consider the air (the mass of air that will participate will be very difficult to figure out) as a part of your system along with those two cars. 
A: @Adam what do you mean by "what is left in the system". 
Consider this example, It will help you think:
Consider an astronout floating (at rest wrt you) in space with a ball in hand. If he throws the ball in a direction. Do you know what will happen?
The astronout applied force F on the ball and thus the ball also applied force F on him by using newtons 3rd law.Thus the ball and astronout if considered as a system will have zero acceleration as two internal forces cancel out to give a net zero force on system. But if only ball or only astronaut is a system then they have the Net external force F acting on them. Thus astronout and ball will have to move.
 The direction of astronout's motion is in oppossite direction in which ball is thrown. They move in such a way that their centre of mass is always at rest. But individually they are in motion as they had an net external force applied on them.
*The scene was set in space to avoid other effects like friction etc. 
*The  internal forces(due to third law they cancel in pairs) can't produce net force on a system and thus  no force means no change in momentum and thus momentum is conserved in absense of external forces.This is conservation of momentum principle. 
A: How did physics end up with momentum and energy conservation?
By observing, experimenting and cogitating over the results of kinematic experiments the laws of conservation of energy momentum and angular momentum were discovered to hold. The whole mathematical model of Classical Mechanics, is successful and has these laws inherent in the formulation BECAUSE this is what has been observed and measured over and over again.
When building up a thought experiment , it is not wise to question basic laws in the framework where they were derived. One does not have to think "why" , about conservation laws in their appropriate frame , but "where am I going wrong in this thought experiment".

Two cars with the same mass and different velocities, say −5i and 10i, collide and no external forces act.

Let us suppose they have the same mass for simplicity, then their center of mass moves with 5i/2. 
Conservation of momentum means that the mess after collision moves with  5i/2 velocity and the momentum will be 5i(2M)
Of course it finally comes to rest due mainly to frictional forces that stop the slide ( or other objects scattered in the way).

All their kinetic energy goes into sound, heat, and deformation. So final velocity is reduced to zero for both of them. 

Some will go to sound heat and deformation, and part of the momentum will be transferred to atoms and finally the earth, but a lot of the 5i(2M) will remain for the mess to move after scatter and do extra damage.
