The problem is clearly visible at maximum vertical sensitivity (starting from 5mV/div.). When the second channel is turned on, both beams move upwards. To return the beams to "Zero", you need to switch both channels to the "Ground" mode and back, then both beams are combined into one "zero" line. Many people have this problem, maybe you are lucky.
Yes, and at the same time, a relay clicks on the second channel. There is one more point: when the second channel is turned off, the voltmeter readings are absent, and in the first channel, when it is turned off, they remain the same as they were before the turn off.