How photons can emerge quantized if their cause is continuous? I know that photons are quantized, they are not continuous. But they are created by an accelerated charge. So how is it possible to have a quantized outcome from a symmetric continuous event? I mean what decides the direction and the moment a photon is released if its not being continuously released in all possible directions with appropriate magnitude?
I cant imagine a law like, an accelerated charge just sends off photons in a random direction at a random time. Isn't this a bit weird to have quantized particles emerge from a symmetrical event? (as long as they are quantized, they are obviously non-symmetric wrt. source)
 A: 
So how is it possible to have a quantized outcome from a symmetric continuous event?

Easily. So easily that I'll describe the easiest example to me. Which is to describe what happens when a Stern-Gerlach device interacts with a spin 1/2 particle.
You could have a particle with any spin whatsoever, but no matter what single particle state you pick it will be eigen (with a positive eigenvalue) to $n_x\hat\sigma_x+n_y\hat\sigma_y+n_z\hat\sigma_z$ for some real unit vector $(n_x,n_y,n_z),$ so let's say it is eigen to $\hat\sigma_x.$ Then we send it into a Stern-Gerlach device oriented in the $\hat z$ direction.
We send it in as a beam travelling in the $\hat y$ direction, it widens in the $\pm\hat z$ direction and then splits into two. Meanwhile, since the spin state is always eigen (with a positive eigenvalue) to $n_x\hat\sigma_x+n_y\hat\sigma_y+n_z\hat\sigma_z$ for some real unit vector $(n_x,n_y,n_z)$ we can describe how that vector changes to describe how the spin state changes.
What does it do? It is a vector at every point (just as the spatial wave is a complex number at every point) and over time as the beam widens and splits the vector rotates. It rotates to point in the $\hat z$ direction for one branch of the Y and rotates to point in the $-\hat z$ direction for the other branch of the Y.
You had a beam with a spin, and the beam continuously split into two and the spin continuously changed from one result into a pair of results, one for each branch.  Is this made up? No, a Stern-Gerlach device you make in the lab has an inhomogeneous magnetic field, and I just literally described what the Schrödinger equation predicts for the evolution of a spin 1/2 particle in an inhomogeneous magnetic field. So that is what happens.
Two discrete results (a spin vector pointing in the $+\hat z$ direction and a spin vector pointing in the $-\hat z$ direction formed from a symmetric continuous evolution.
What happens is the state evolves so that new properties emerge ($\pm\hat z$ spins) that didn't exist before but that are correlated with other properties and these correlations allow new connections with new things. When enough new things are connected to one result versus the other result to the point where each branch acts like it is the only branch, then you can introduce an assymetry, the assymetry you'd get if you were a single celled organism and you split in two and the two new halves wandered off and never saw each other again and then just assumed your other half was never really real. If you will never ever influence each other, even indirectly, there isn't anything scientifically wrong about assuming that, from the perspective of each of you it is even efficient to do so.

I mean what decides the direction and the moment a photon is released if its not being continuously released in all possible directions with appropriate magnitude?

If you assume there is a photon that is born and travels and then interacts, then you will run into problems at some point because that story is provably wrong. It's not that there are photons out there and we just don't know where they are and they do random things.
You have a universe with parts and the parts sometimes interact and sometimes don't. You can make new things but when they interact they can interact by splitting the universe into parts that develop to become unseat and incapable of interacting with each other.
Then, since each part sees just a fraction of the possible outcomes, it can look, from their own parochial perspective that something assymmetric happened.
A: 
I know that photons are quantized, they are not continuous.

Photons are not quantised, nor are they continuous. They are the charge carriers of the electromagnetic field as arising in quantum field theory. An accelerated charge generates an electromagnetic field whose carriers are, in turn, the photons whose energy might be quantised.

So how is it possible to have a quantized outcome from a symmetric continuous event

There is no constraint why it may not be so. An outcome, as per in quantum mechanics, is one amongst the set of eigenvectors of an operator on a Hilbert space, and being this set countable or not only depends on the mathematical form of such operator. There is an entire branch of mathematics dealing with spectral theory of operators on Hilbert spaces and all the answers can be found therein. Any general (and continuous) physical system is described by means of some operators, which are allowed to have discrete spectrum nevertheless (this usually happens whenever the states of the systems are bound states, namely the state can be located everywhere in within a bounded spatial region).

I mean what decides the direction and the moment a photon is released if its not being continuously released in all possible directions with appropriate magnitude?

Just write down and solve the emission equations for an accelerated particle: there is plenty of literature thereon (any textbook in quantum mechanics or even classical electrodynamics).

Isn't this a bit weird to have quantized particles emerge from a symmetrical event? (as long as they are quantized, they are obviously non-symmetric wrt. source)

You seem to have a somewhat odd idea of what quantisation means. It has nothing at all to do with symmetries and continuous motion, rather it means that for some reasons observable quantities have discrete (namely, quantised) values. This being weird or not, it is just the way the universe works: physics (and science in general) does not describe why it is so, rather it describes how to predict measurements through mathematical equations.
