Why do we need to define things like angular momentum, torque, etc.? I want to do everything with fundamental quantities. The conservation of angular momentum relies on the conservation of linear momentum. Torque is similar to force. For example, is it possible to say whether an object will rotate without doing the $\mathbf{F} \times \mathbf{r}$ multiplication? If not, why does this multiplication provide us with such information?
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The fundamental reason is that momentum is not simply a vector (a quantity with direction and magnitude) but also has a line of action. A force, which is the rate of change of momentum, likewise. So the effect a force has on a body depends where you push on it. Push on the left side, and it accelerates clockwise, push on the right side, and it accelerates anticlockwise, push in the middle and its angular velocity doesn't change.
We can represent this mathematically using projective geometry. We add an extra dimension to our normal space (let's call it the '$w$' axis, alongside $x$, $y$, and $z$), and place the origin a unit distance away from it. Straight lines through the origin intersect normal space in a point, planes through the origin intersect normal space in a line, and so on. This way, we can represent vectors through any arbitrary point using planes through the origin. As a bonus, we have vectors and planes parallel to the normal space that can be used to represent points and lines at infinity.
This is the same trick as using homogeneous coordinates, where we add a $1$ to the end of a vector, and find we can now represent both translations and rotations about any arbitrary point in a unified way using a $4\times 4$ matrix. In this picture, a translation is simply a rotation about an axis parallel to normal space - i.e. a rotation about an axis at infinity. We can see this if we imagine the rotation axis being moved further and further away - it starts to look more and more like a translation.
This 4D-plane representation captures the full behaviour of momentum and force. A plane in this 4D space can be decomposed into six components, one for each pair of coordinates. These are $wx$, $wy$, $wz$, $yz$, $xz$, and $xy$. The first three are the linear momentum, their rate of change is the force. The last three are the angular momentum, with torque as the rate of change. (The mass also turns out to be 6-dimensional, with the first three components identical (in flat space) and equal to the normal mass, the last three components being the moment of inertia.)
The "fundamental quantity" here is 6-dimensional - called a screw - and represents a plane in a higher-dimensional space intersecting with our normal world along a general line: one not required to pass through the origin. But vector algebra can't cope with such a fantastical beast, so we split it into two parts that we can handle, one of them acting as a vector (linear momentum or force), and the other as a pseudovector (angular momentum or torque).
The interesting case is when the force plane doesn't intersect the normal world at all, but runs parallel to it. In this case, we have a pure torque, with zero force. The special case of a torque with no force is called a 'couple'. We still require conservation of angular momentum for varying torques, even though force is always zero. (And hence $r\times F$ is $\infty \times 0$, and so indeterminate.)
For more information, look up screw theory. For example, there's quite a nice approach here using Conformal Geometric Algebra to simplify the equations of rigid body mechanics (page 17), although CGA is a bit of an acquired taste.
