Bxdty

As far as I understand Scales are defined on intervals. So, for example, the major scale is defined by the following intervals from the root:

C - Root

D - Major Second

E - Major Third

F - Perfect Fourth

G - Perfect Fifth

A - Major Sixth

B - Major Seventh 

This is perfectly clear to me, and it makes sense, because we have 7 note names, so every note must be included on the scale, but then when trying to figure out the proper name notes for an octatonic scale it becomes blurry to me. I mean when building an octatonic scale you only stack Tone-Semitone-Tone-Semitone...

But how do I properly define the Note names??

For the semitone spacing the interval could either be a augmented prime, a minor second, or even a doubly diminished third.

For the tone spacing the interval could be a doubly augmented prime, a major second or a diminished third.

How do I find the proper note names for non heptatonic scales?
Does the same rules apply for the chromatic scale?

In heptatonic scales, we have one instance of each note name, because there are seven note names. But you're correct: if the scale doesn't have seven tones, some note names will need to be repeated or omitted.

Octatonic ("diminished") scales have been addressed in Notating the diminished Scale

The same stipulations discussed in that question apply in other non-heptatonic scales: some composers prefer to write exclusively with flats or sharps, and composers often aid our understanding of a tonal center by how pitches are spelled.

But there's one additional stipulation that I can think of that applies to your question. Let's imagine we're writing a hexatonic scale, often understood as a half-step pattern of 3 1 3 1 3 1:

This looks like a great way to notate this, and it is: we have no repeated note names, and we only omit the F. This is what mathematicians might call a "maximally even" distribution of note names, meaning it's as evenly spread as possible, considering one note name must be missing.

But let's look closer at this collection. It contains the following triads:

CM

Cm

EM

Em

A♭M

G♯m

Sometimes, composers writing hexatonically like to highlight the juxtaposition of two harmonies. And if they want to highlight the mixture of, say, the C minor and E major triads (what we call hexatonic poles, because combined they create a hexatonic collection), a composer may well choose to notate this collection as:

Even though the two versions of E and G look awful, the notation makes clear the juxtaposition between the C minor and E major triads.

In short: depending on the musical situation, some composers (though not all!) may choose to use notation that looks less than ideal because it illustrates the musical process more clearly.