"From what I understand so far, photon creation is a result of an electron oscillating around between energy levels in an atom."
It's a little more correct to say that an accelerating charge creates an electromagnetic wave (it radiates). First start with a static picture. Imagine you have a single electron in your hand. The electron creates an electric field: if you place another charge in the vicinity it will experience a force (or potential) based on the presence of the electron in your hand. http://physics.bu.edu/~duffy/PY106/Electricfield.html
At a point 1 meter directly to the right of your hand, the electric field has a given value. Call this point $q$.
Now imagine you move your hand up a meter. Some time later, the electric field at point $q$ changes compared to what it was when your hand was 1 meter lower. Now here's the key: the electric field can't change instantaneously 1 meter away from your hand as you move your hand. The change in the field can only travel as fast as the speed of light. So in the first instant where you are moving your hand up, the field a meter away still looks like you haven't moved at all. There is a delay between you moving your hand and the field changing to reflect that some distance away.
You can imagine the effect of this delay as a kind of 'kink' in the electric field generated by the electron in your hand.
Now imagine you start wiggling your hand up and down. The single 'kink' becomes a set of ripples which will keep being generated and - hooray! - you now have an electromagnetic wave propagating through space.
Now imagine a place really far away from your hand. When you first start wiggling your hand, the person really far away won't see anything. After some time (to be precise, $t=d/c$, the person will see an EM wave "train" begin to pass by.
What happens if you stop waving your hand? The wave stops being produced. The far away person sees the end of the wave "train" some time later.
If you only wiggle your hand for a little bit, the wave "train" will be short. If you wiggle it a very tiny bit, then you approach the limit where the wave train becomes more like a wave packet, and eventually, a photon.
Leaving photons aside for a moment and going back to the wave picture; you might find it illuminating to know that "wiggling" electrons is indeed how we create electromagnetic waves. Wiggling the electrons up and down in an antenna using an oscillating voltage source is roughly how we create radio waves. Wiggling the electrons back and forth faster and faster is exactly how we create higher frequency radiation. So:
"Does it work the other way around with photon production, that is, do faster oscillating electrons create higher energy photons, which then can go and excite other electrons?"
Yes! Your instinct is good. But it's important that the charges are accelerating.
There's one small misunderstanding to clear up, however:
"Higher frequency photons (and I was struggling to grasp what that truly meant, and if the answer to this question is yes, I think I have finally understood it) hitting electrons in an atom gives the electrons more kinetic energy, that is, makes them move faster."
A high frequency photon has a large energy. $E=hf$. So if an atom can absorb it, the electron will be promoted to a higher energy level inside that atom. It's incorrect to ascribe this completely to kinetic energy. The energy of a given atomic orbital is determined by the angular momentum, the potential energy with respect to the other charges in the atom, and so on.