I had strange constant noise when using my rear USB3 ports. Switched to the USB2 ports and the noise stopped. There have been several updates to the USB3 driver since then but I haven't tried it out. Easy enough to check if that  is your problem it seems.

Even with RX 5 Adv, it's very difficult to clean the noises, there're all kinds: humm, buzz, cliping, distortion, radio interference...

> The signal ONLY happens when the sound card is being accessed (by a program),
> and ONLY if at the same time the drum module is plugged in (to the desk).
 
It's a switching supply ground loop between those devices.
Filtering will yield disappointing results. Fix the ground loop.
 
Edit: If lifting the ground makes you nervous (as it should), then get an isolating
transformer and remove the ground loop on the signal side.
 
I get good results with the Behringer MicroHD HD400. I have several.
Connect the cables from the drum module to it, and then connect the
outputs to your desk.
 

You only mention "plugged into the desk"  This implies an audio cable?? the power supply? 
Have you tried disconecting the USB cable? 
 
I get a crackling noise in my system , only when the audio engin is running, just like you are discribing. Every move you make with a mouse causes a buzzing too. 
I slowly eliminated each component and swapped out gear and I isolated it to my old Yamaha P 2100 Power amp. Go figure. Remove the power amp from the chain , crackling gone. 
 
But my point is you need to isolate each component.  The Drum module has a headphone output ,, right? try that in stand alone. 
Then hook up the USB cable..
Then try a different computer...on and on 
 

Any mobile phone sitting near the drum module or on the desk?

I had similar noise and I believe it was due to a ground loop picking up digital signals like USB packets, or other high frequency digital signals causing modulation into the audible spectrum (like AM). You can often change the sound by just moving your mouse around for instance.
 
As suggested by jpeterson and others, the best solution is probably to ground lift the audio signals using something like the Behringer MicroHD HD400 ($25 on Amazon). In my case it completely solved the problem.

Hey, I didn't get notified about the last few replies for some reason.
 
I'm really on a deadline and can't afford to try any further things (and I had/have already done
all the tests recommended above). - Except the hardware hum-filter.
 
The solution I had in mind was using the LP-64 Equalizer (which I could use to match the spectral pattern in reverse). I thought that I would be able to adjust the wet/dry levels of it however it only has a master output knob.
It's going to be hard to predict how intensely to apply it as I'm layering 11 different drum layers (which I'm tracking separately for total control).
 
If anyone knows either an alternate (similar) EQ I could use (with a wet/dry option), or how to use this still I would
appreciate it. - Could it be done with some kind of send setup?  ***I have never used send knobs in a production
and wouldn't know how to apply it here.
 
 
Thanks.

You might try the trial version of Adobe Audition. Audition has pretty good noise reduction/removal, especially if used moderately for several passes over the audio rather than trying to remove all the noise in one go. I might be wrong but I think the trial version is fully featured until it expires after 30 fays.

You can also try iZotope RX5.  I couldn't do much with the files you've made available but I think you can have very nice results with the original files.

That noise will ruin any sound so you need to track it down.  Have you unplugged EVERYTHING and added things back one by one.  This includes the power.   I've had just about every type of noise at one time or another and that strategy ALWAYS tracks down the culprit.   Its not a trivial operation but in the end, you also get the opportunity to unscramble what has likely become a mess of cables.  Also, being digital, do you need to sync it in any way?  I would say it is related to power and is exacerbated when you plug in the drum module.   Again, unplug EVERYTHING, then reconnect the most basic stuff like interface and monitor so you can hear and then start from there.  Maybe go withthe drums next as they seem to be triggering the issue and then one by one, power first , then audio connection (test each separately)