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Yeah, it's often the little and unexpected thing that is responsible for all the problems :D

Well, I'll try to connect PSU's ground to GND of the DRV to see if anything changes. Will edit soon.

Edit: No change. After connecting PSU's GND to DRV's GND by cable, same things were happening.
Well, changing DRVs and trying them on another printer, or simpler - trying A4988 on the toyrep - seems to me like the fastest and the only reliable way to find out if the RAMPS or DRV is bad (or neither of them).