On retreat I started to really strongly notice Kundalini for the first time. I started noticing that the kudalini flow would get disrupted by my in-breaths which, as I started to focus on it, still seemed to be controlled / voluntary.

I started to drop all effort to breathe with the expectation that it would just shift to automatic. However, that's not what happened, breathing just stopped alltogether which freaked me out and caused a strong inhale.

Slowly overcoming the fear and allowing the breath to calm down more and more the diaphragm-based breathing almost stopped. Then, instead of any type of breathing I ever experienced, the left and right lungs would start to alternitively move in an up-and-down left-and-right swaying movement. Much more like massaging air in a circular way from the left and right lungs. This coincides with a swaying of energy between the left and right side of the body.

I can remain in this style of breathing for quite some time but fear still arises causing minimal movement of the diaphragm. I'm not sure if that is what keeps me from suffocating or if the rhytmic left-right breathing is actually good enough and my body just freaks out because it thinks that that's not breathing and wants to do a strong inhale.

Thus my question is if this is actually a type of OK breathing or am I confusing the kundalini flow from the left and right body sides with movement of the lungs, or maybe the lungs just naturally move and I never noticed but it has nothing to do with breathing?

Is it expected that the inhale stops if you drop the intention to inhale? How do I let go of controlling the breath if I have to control the moment of the inhale?

Why does the kundalini flow get disrupted on initiating the in-breath?

After writing the post I think I understand that what I described as diaphgram breathing is actually chest breathing, so maybe that is the problem and I need to actually do belly breathing instead but fear is causing chest breathing.

You will be fine. The automatic breathing is probably more natural than your normal patterns anyway and the body is just returning to its best performance. The fear is something of a mental barrier so I suggest you only further investigate it in order to squash this phenomenon. 
A byproduct of meditation is the different forms of practice that oxidize the body. Its a more subtle aspect to touch on. But you are actively changing the chemical composition in your system. Stagnant air is being cycled out and more cells replenish their stores of oxygen. "You" will be required to breathe less. 
If you are approaching this with the intention to learn, then you can extract information from even the most fearful experiences. 

The classic insight stages 2-4, meaning Cause and Effect, Three Characteristics, and the A&P, are common times for the sense that attention is interfering with the breath, then that the breath is doing weird things, then very odd patterns of breathing sometimes, as well as Kundalini phenomena, which are at their potential peak in the A&P. You can read more here: https://mctb.org/mctb2/table-of-contents/part-iv-insight/30-the-progress-of-insight/

We don't know why these happen like they do, but they have been observed happening like this in tens of thousands of practitioners for a very long time.

Best wishes!

Daniel

maybe you are starting to be more aware of your breathing because of the practice and being more active in controlling your breathing consciously

this then cause fear when you do not control it, you will not be able to breathe, am I guessing correctly here?

Breathing balance is important and both nostrils must be able to blow the same power of air everytime you breathe out, in order to have a balance body

One sided blocked nostril means that your Ida and Pingala is imbalance which also means that your kundalini can't go up using middle way or shusumna

And one reason for having one sided blocked nostril in men is mostly cause by excessive masturbation

Thanks for the answers everyone, it helped a lot. To follow up with has happened since I posted: it feels like my default breathing has shifted from chest to belly when I'm at rest or in meditation, and that the weird "stopping" of the breath was the transition period. Now, the breath is very calm and feels completely automatic again, there is no fear in relation to it, and overall it feels more wholesome/healing/centering.