In elementary particles all particles that have spin different than 0, spin, i.e. have angular momentum, so photons are spinning too, they have spin 1. There exist particles and systems with spin 0 (pions as an example), those do not spin :) .
Since physics started from macroscopic studies one has to look at the equations that describe motion classically, the solutions fit data perfectly. These equations obey "Noether's theorem" that shows there are conserved quantities in the dynamics of motion coming from the symmetries of the system. Energy, momentum and angular momentum are conserved.
This means that once a path or a system rotation is established by some interaction, for example by a grazing impact of two asteroids, if there are no further interactions the asteroids will keep on spinning because the angular momentum they gave each other will be conserved individually.
So the answer to your question
Where has all the angular momentum come from? Why is it so natural?
is : from conservation laws. It is natural because equations of motion and conservation laws are a description of the mechanics of nature, and that is the way nature works.
Now space time and angular momentum are another story in General Relativity, where, because a rotating object has acceleration in the radial direction it distorts space time around it.
Edit after comment:
The original energy that set the universe in motion, created the particles and induced rotations is described currently by the Big Bang model, at the origin of our universe billions of years ago, starting with quantum mechanical fluctuations.