Which brings me to my question, how DO you write successive, non-functional chord progressions like chromatic ascending/descending bass?

You write them using pencil and paper, sometimes a computer, sometimes just in your head, sometimes as you play an instrument. I believe your real question is how do you create successive non-functional [sequences]? Keep in mind that the use of the word "progression" implies a certain functionality to your chord choices. Also keep in mind the word "functional" as you are using it describes typical tertian voices, smooth voice-leading, and the anticipation and resolution of harmonic cadences.

Here is one exercise: pick a note on a piano, and reharmonize the note as many times as possible - say we choose the note "E". Here are some possible chords that may be harmonized with E:

• Emaj, Emin, E+, Edim, Cmaj, C+, C#m7, Fmaj7, Ab+, Bbm7#11, etc etc

As you can see, through simple reharmonization we can recontextualize any not to any desired result / effect. Therefore, to write a sequence including a downward / upward chromatic bass line (or any line for that matter), the bass notes may simply be reharmonized to be contextual with different chords; how different or similar the chords are to one another in tonal region is a matter of the composer's whims. The added dimension of bass lines is that you can also reharmonize with inversions as well. For example, the Bbm7#11 chord from above could have an E in the bass, thus putting the chord in 5th inversion.

For related reading, check out my answer to this question about chromatically moving bass lines.

Do you play on the keyboard and hope something good will arise through trial and error?

Sometimes, it depends on how the composer writes. Sometimes you stumble across something unknown to you, and sometimes you need it work out an idea that came to you in abstraction. Composers are obsessed with working out puzzles.

Are there any rules or advanced music theory topics regarding how to write it?

It really depends on your definition of function. For example, in Set Theory, you would use your sets (usually hexachords) and intervals to derive your chord-language. In Serialism, you use the order of the tone row for your pitch-language. In either of these scenarios, aural connections defined as "progressions" may not be readily heard / perceived, but they do succeed in unifying a work.

If the function of chords in your music is to creating a terrifying sound wall or a barely audible whisper, then your method of composition should reflect your larger goals.

Music is unified through repetition. If nothing in your music repeats (no chords repeated), then a particular, un-unifying effect is created, and therefore the chords truly have no function (unless it was your goal, then of course that would be their function).

Does it just come by naturally during the process of improvisation? Or is it a talent that someone is just good at without reason?

These questions relate more to how you think other people create "non-functional" progressions. Yes, some people may hear chords a certain way, as others have put it, if it sounds good to you, it probably is. However, as I've shown above, you can use simple theoretical knowledge to greatly diversify your chord-vocabulary.

I feel it is important to point out that many contemporary bands / artists that are not trained stumble into certain sequences through ignorance. They are unaware of how the chords are supposed to be put together, and so therefore many songs result from a lack of foundational knowledge - progressions using "beginner" chords run rampant: Am, E, G, D, C, F all just sort of thrown together in any four-chord order.

I would not posit that someone who comes up with a "non-functional" progression is "talented" as anyone through ignorance can play cluster chords on a piano (much like any non-painter can throw paint at a canvas). Rather, it is how the chords are used that makes the difference. If someone is stringing chords together because they don't know any better, it will create a certain type of music. If someone is carefully planning how their chords interact, then again, it will create a certain type of music.