One last test for today, it's already late here...
Accordingly, I was lazy and controlled the scope from the living room.

The web connection is fast, very good.
Noise level measurement:One channel (CH1, the others behave in the same way), 1 mV per division, 1 ms per division.
Full bandwidth, 20 MHz, I skipped the 250 MHz.
Then 50 ohm input and 1M ohm.
I consider around 124µV to be good for a 1GHz scope(yes it is...

) ; at 1M ohm, the level is lower, which is probably an indication that the bandwidth is more limited at 1M ohm—I will test this soon.
*Then I also used Hi-Res Mode, which significantly reduced the level again, but that's no surprise, as the bandwidth drops to 2.5kHz at 16 bits.
Finally, an FFT up to 1Ghz, in the new Averaging Mode (20x).
I discovered something else that wasn't very nice.
At 100ms/div, it's obvious that you can't display an FFT up to 1Ghz or even more – yet you can still enter a “max” span of 2Ghz in the menu.
This shouldn't be possible; “max” should refer to the current FFT sample rate.
I found similar nonsense in the AWG menu, where there are buttons for kHz, MHz- and GHz...
EDIT:
*Those who can read a data sheet have an advantage:
500 MHz bandwidth at 1 MΩ input.