Do our ears (or brain) only perceive periodic disturbances? Through a little time spent using sound editors for this and that, I've come to notice that if I play any musical note 440 times per second, I hear an A, whose frequency is 440Hz. Then I tried with very non-periodic sounds, even long ones like an entire song, sped them up and played them repeatedly at 440Hz and it worked. Not that surprising, but a little confusing. Confusing because first of all, any periodic wave has an infinite number of periods : possibly a smallest one, and then all its multiples. So which ones do we hear ? I'm guessing the notion of harmonics has something to do in there (which I am familiar with as a musician though I never had any true understanding of them). Confusing also since indeed, any non-periodic wave put together a few times becomes periodic, even a musical note followed by a silence, repeated, has a period equal to the duration of the note added to that of the silence.
Now finally here's my actual question : when we hear a note consisting of a random sound repeated at high frequency, what happens if we hear it during only one period, and if that one period itself is not made of a periodic signal ?
I can reformulate more mathematically : if the air pressure at the point my ear is is the function $\sin(2 \pi \cdot 440t)$ I hear an A, but what happens if I hear it for $\frac{1}{440}$ of a second ?
My own thought experiments incline me to say we don't perceive it, and formulate the theory that we only ever hear anything that has a frequency. In other words if I played a guitar string and could mute it the first time it came back to its resting position, then I wouldn't hear it.
Otherwise it would make no sense not to hear something lower than the human threshold of 20Hz just because it has a frequency and hear something else that has no frequency.
 A: What you perceive as sound is just your brain's  interpretation of pressure changes in the inner ear behind the ear drum. When an air disturbance makes the ear drum move causing the pressure at the other side to change, your brain "hears" it.
Any motion of the ear drum is in this way "heard" as some "sound". Wave or not and periodic or not. 
But you might not want to call it sound. You would rather call it noise. If you have ever been standing in a "silent" room (with all these pointy foam spikes on the walls), you'll realize that you in your daily life constantly hear noise. There is constantly some disturbance of your ear drum. 
If you happen to hear a periodic disturbance, it will sound more ordered. It will sound as actual sound and not noise. If you hear just one period during one fourhundredandfourtieth of a second, then you can't in any way recognize it or distinguish it and you would call it noise along with all the other random disturbances in the air pressure. 
Defining if what you hear is sound or noise depends on the person, and I don't think there is a specific official definition stating if a specific amount of period is required in order to call it sound. The degree of orderedness and enough periods to notice it as a periodic disturbance is subjective. 
A: Our hearing is primarily frequency based.  In particular, our brains typically structure their concept of a "sound" around a so-called "fundamental frequency."  Sounds can be decomposed into many frequencies which are often organized as harmonics.  The lowest tone in this harmonic series is the "fundamental" frequency of that sound.  Our brains then proscribe character to that sound based on the overtones.
There are a few places where this can break down.  One great example is in Tuvan throat singing.  In this art, one sings a drone and then manipulates the overtones with the shape of one's mouth to highlight a single overtone.  When we hear this with our ears, we hear the normal harmonics of the done and think "that's one voice," but the highlighted overtone is too loud to fit in that distribution.  It's much louder than the nearby harmonics.  Thus, our brain chooses to treat it as "a second voice," and we get the illusion that Tuvan throat singers are actually singing two different notes at the same time!
We also capture some transient non-frequency information for purposes of spatial location.  If I clap from one direction, you can tell where the clap came from because you're measuring the difference in timing between the signal reaching your right ear and the same signal reaching the left.  We can show that this information is essential with what I consider to be a hilarious test setup.  Put someone in a room with hard walls with two speakers, we'll call them A and B.  You start playing a tone out of speaker A (any tone, midrange is best).  Initially your ears will hear the transient effects of this speaker coming alive and you'll associate the sound to speaker A.  However, with all those hard walls, pretty soon you'll loose track of those transients.  All you'll have to work with is frequency data.  At this point, we shift the sound from A to B, varying the power using an exponential curve.  This transition makes it very hard for the ear to detect that speaker A is now silent and speaker B is now playing.  Thus, you'll actually hear the sound come from speaker A.  This effect can be so disconcerting that, even when you are told you are listening to speaker B it can be impossible to convince yourself.  It is apparently quite disturbing to have someone tell you this, then walk up and unplug speaker A.  You'll keep hearing the sound out of speaker A, even though it is unplugged!  Magic trick!
So we do have tools in our wonderful squishyware brain to recognize facets of audio other than just frequency.  However, evolutionarily speaking, the signals which could be well captured by frequencies have been very important, so the majority of our processing uses them.
A: Regarding the question in the title: Do we only hear sounds with frequency?
Audible sounds have (waveforms in) frequencies in approximately the range of 20 Hz to 20000 Hz. Sounds that are perceived to have a dominant frequency/pitch are called tones.
To quote Wikipedia: Pitch is a perceptual property of sounds that allows their ordering on a frequency-related scale.
We say that a sound has a certain pitch because it largely has the properties of a sine wave of the associated frequency. That is, there is a dominant frequency and it is the one we associate with that pitch.

if I play any musical note 440 times per second, I hear an A, whose frequency is 440Hz.

When you took a tone with a certain pitch and sped it up to play in the frequency of A you simply got an A tone (whose deviations from the ideal sine function were determined by the original tone). Because tone is determined by its frequency/pitch.

...what happens if we hear it during only one period, and if that one period itself is not made of a periodic signal ?

This confuses me, hopefully you could improve that part of your question now.
You would probably be interested in these:


*

*https://music.stackexchange.com/questions/3849/how-do-harmonics-work

*https://sound.stackexchange.com/questions/28163/whats-the-shortest-sound-perceptible-to-the-human-ear

*And maybe the sample chapters here: http://whyyouhearwhatyouhear.com/subpages/sample.html
I hope I did not say something incorrect ;)
A: 
if the air pressure at the point my ear is is the function $\sin⁡(2π⋅440t)$ I hear an A, but what happens if I hear it for $\frac{1}{440}$ of a second ?

If you have a "sufficiently long" sample, then the frequency becomes well-defined.  You can imagine doing a Fourier transform on the sample and you'll get a nice peak at the dominant frequency ($440 \text{Hz}$ in your example).  
As you shrink the sample, the uncertainty in the frequency increases.  A single "pulse" doesn't have any periodic content.  If you take the transform, you'll get a very wide frequency output.  The harmonics created by going from zero output, to your signal, then back to zero in a short time can excite a range of different frequency sensors (such as the ones in your ears).
As an example, try taking a second or two of a pure tone, then make it smaller in an audio program like audacity.  As you make it shorter and shorter, the actual tone of the sound is lost.  The sudden start and stop overwhelms the tone and makes it sound like noise.  This introduces a signal that is not present in your $20\text{Hz}$ tone and allows it to be perceived.
