I just saw some of your replies, thank you for taking the time of helping me.
As you said, one of the problems with the scope was a bad solder connection going to the CRT. Reflowing that solder joint fixed the problem of the screen fading away (or at least it hasn't happened since I fixed the joint).
The only problem remaining is the bright spot on the left of the trace.
Now, I've noticed some stuff about those bright points now that I can keep the trace for a longer time, so here's my theory about it:
What I've noticed is that, changing the time base, sometimes a ghost trace appears, and it seems to be a zoomed version of the waveform in display. This only happens when the time base is at 1 ms for the square wave, but in all the positions for a sine wave I've used to test it.
You can see it in the attached pic (btw, the trace is bright, but not as bright as it seems in that pic, the camera seems to be sensitive to that color).
After taking a look at the schematic posted and some of the user manuals, I've noticed that the "COS 5041" version of the scope has a delayed time base for waveform magnification, with its own knob, which is not in the 5040. Also, the PCB seems to be the same, and they populate or don't populate some of the components.
So what I'm thinking is that maybe some of the components in charge of that waveform magnification are present in the PCB of the 5040, and for some reason the magnification is turned on, but since there's no controls in the front panel, the magnified part is always just the left part of the waveform, which would explain the brightness in that side, and the second waveform visible.
It could be any other number of reasons, I've been a bit bussy and tomorrow I'll be able to dissasembly the scope to check the whole circuit, since it envolves taking all the knobs and a lot of work with the connectors etc.