...now, the weird part is: it's not all the strings, not all the time, and the problem seems to evolve or whatever.

Basically it started some time ago (a month ago or so) with just the B (or A# to be precise - I tune half step down) string buzzing randomly when not fretted, more recently the G (or F#) joined in although not as audible, and now it seems the low E(/D#) is joining the party (that's right, never happens with the A and D strings which are in between). The sound is quite weird, it's not the same kind of buzz as with too low action/incorrectly adjusted truss rod kind of thing, it's more of a "hollow" sounding buzz.

I don't really have an explanation for this of any kind - it seems to be the buzz is coming from around the nut area, but I'm 95% sure the nut is perfectly fine - I'm pretty sure if it was badly cut or anything similar, the problem would have happened since the beginning, not gradually after some time. It's 101% not anything to do with buzzing on the frets, besides that the guitar plays very well. And the weirdest part is, as I said, is that it's kind of random - it appears only every once in a while, and I don't see any real pattern when. Of course it's a lot louder when picking harder.

I've had the guitar for 2 months and literally everything about it is awesome, but this is slightly annoying, and I'm honestly out of ideas (not that I had many to begin with). It has a Tune-o-matic bridge, and as previously mentioned, I tune half step down/D#/Eb with 10-46 strings. It's generally setup well, even intonation is virtually perfect. I did change strings in the meantime, the B string buzz happened already with the old set.

I'd be very grateful for any ideas what could it be, and if there's any more information that could help, ask away.
Check if the nut is any good by fretting each string at the 3rd fret, look that there's plenty of clearance at fret 1.

If that's OK, check the neck to see how straight it is.
I'd start with a complete setup beginning with the neck relief and new strings.

Also check the clearance at the first fret (proper nut fitting) and check for high frets.
Ibanez RG1570 Prestige
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Jackson DKMGT Dinky (EMG 81/85)
ESP E-II Eclipse Custom (JB/'59)
ESP LTD EC-1001FR (EMG 81/60)
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TOOOO many T.C. Electronic Pedals. . .
I'm 99% sure the sound does not come from the strings touching any frets - there's plenty of space between the strings and the frets. Also, I did adjust the neck relief a little, it's generally fine - I think the source of this is elsewhere.
Either they touch the frets or don't. If they don't and the nut is alright that leaves you with the saddles, or if you're getting this while plugged in some electronic issue.
If you're sure this is not fret buzz, you need to start looking at other things. I've seen loose parts vibrate, causing buzz that is not easy to find...tuners, bridge parts, pickup switches, output jack, pickups, and in one case a truss rod that had no tension would vibrate inside the neck. Just a slight tightening fixed that, it had a little tension and couldn't vibrate any more. Anything that is held in with screws is suspect. Does the nut itself move? Any loose parts can create a buzz. Start checking anything and everything. If the strings are not buzzing against the frets, something is probably loose. It could be anywhere...
Hmmm...I wonder what this button does...