Cell phone was not far away, but it was not the reason for my problems.
After another game of lets-see-how-many-ways-there-are-to-plug-things this morning, I think I have narrowed down on causes.
First, was CEntrance DACMini. Yes, really. Somehow it injects noise back into computer's USB, and 9xx is picking it up. Isn't that weird? Like it was trying to get competitor out of my system
Anyway, that was the cause for the 'dog whistle'. Unplugged for now.
Second, was powered USB hub that I was using - it was adding distinct 'usb' (random frequent clicks) noise. Getting rid of it and plugging directly took care of clicking.
Third was using USB sockets in front of my computer. They are connected inside by 'usb extenders' wires and were picking up/injecting 60Hz hum. Plugging to USB socket on the back of my computer solved that problem.
So, it works much better now. Still, DACMini was immune to all those issues, its USB connection is truly bulletproof, if not a bit unfriendly to other audio devices in the same setup. To be fair, only one another DAC that I had was also this resilient to usb noise - HRT MusicStreamer II, everything else picks it up to some extent, Scarlett 2i4 being the worst.
P.S. For some reason after all those 'changes' gap between sound of 9xx and Gustard H10 (upgraded with pair of AD797) shrunk considerably. Maybe it is just my hearing, its quality fluctuates from day to day, and tends to be better at night. Maybe not. But right now 9xx and H10 sound like siblings. At least they do on DT880 that I have 'volume-matched' right now. Arms race is on, probably will get some Bursons for that H10 later.
P.P.S. Another mystery solved after switching back to HD600 - difference is back, 9xx has deeper/better bass and overall smoother, more transparent, albeit warmer, sound. DT880 are not resolving enough, and 600 Ohm impedance probably pushed both 9xx and H10 into different mode where their performance was surprisingly similar.