Back to scope 2. Instead of getting rid of the thermal paste, I simply reseated the cooler and I think I did a better job this time, because there was a bit of click at the end.
Let's check the temperatures when the thing is running. This scope runs WinXP which supports USB drives out of the box so installing Core Temp is really easy. At rest, I'm getting 40C and 100% running, I get a max temp of 64C.

Not bad!

I consider this problem solved.

Next is the noise. Here's what it looks like:

Someone said he fixed a similar issue by rerunning calibration. I already did run calibration, but only for 1 channel. When that completed, I canceled calibration because it takes very long. So this time, I ran it for the full hour and... tadaaa!

The noise is gone!!!

The scope is now fully working. The only thing that's left to do is check the bandwidth (which should be > 600 MHz) and then upgrade the scope to >1 GHz.

But first measure the before case. I'm feeding in the 10 MHz AUX output of the scope into channel 1. Without a pulse generator, I have to hope that the edges of this output are sharp enough.

Using the 0.35/t_rise formula, a rise time of 481ps gives 729 MHz, well above the spec of 600 MHz.

Let's remove a resistor.

So converting an Agilent 54831B into a 54832B and thus increasing the bandwidth from 600 MHz to 1 GHz requires some serious PCB surgery: you need to remove 1 resistor...

Left: before, right: after.

The resistor mod works!

A rise time of 331 ps down from 481 ps, good for 1.05 GHz.

This is within spec, but on the low side compared to the 1.2 GHz that others have seen. It could just be my sample or it could be due to the rise time of the 10 MHz AUX out port being just that.

@joshua has offered me his Leo Bodnar 60ps pulse generator. I'll test that sometime next week.

@joshua The "About Infiniium" screen now identifies itself as a 54832B.

People do crazy things with this scope: replace the LCD panel from 640x480 to 1024x768, add newer motherboards which requires sheet metal surgery, install fast SSDs, update the OS to Windows 10 etc.

But that doesn't really change the way the scope works: it's already on the latest version of "scope.exe". I'm happy enough with the way it is right now. Let's call it a day...