Sometimes the second channel is hard to use independently,

You have single time base, dual trace. I'm used to seeing dual time base, dual trace.

The idea is to be able to "blow up" say the edge of a signal. it's a bit harder to explain.

You will notice that you have an Invert B, this allows A-B or pseudo-differential measurements.

Pseudo means you need a connection to ground somewhere.

 

Keep the jumper wire between Y1 and Cal 1V.
Set the Mode switch to Alt or Chop.
You ought to see Y2 as a flat horizontal line.

Jumper both Y1 and Y2 at the same time to Cal 1V.
You should see square waves for both Y1 and Y2.

Nothing of this worked. No sign of Y2.
The single thing that showed some implicit presence of Y2, was during Chop mode (only Y1 was wired to Cal)...I saw some kind of a ripple on the displayed line, so after decreasing Time/Cm I saw this (in ALT mode, those gaps aren't shown)

 

You have set the Time/cm too low. Set it to about 10ms. Leave Mode on Chop.
See if the Y2 vertical position knob shifts one of the two lines up and down.

 

You have set the Time/cm too low. Set it to about 10ms. Leave Mode on Chop.
See if the Y2 vertical position knob shifts one of the two lines up and down.

Unfortunately Y2 wasn't observed.

I started checking the voltages of transistors according to the manual, to troubleshoot the scope's circuits.
I encountered some discrepancies, so I wasn't sure
1. if I do it in the right way
2. what should I do when the transistor's measured voltage is not as expected

For example (actual case):

(the manual says that all Y channels should be set to Ground, the amplifiers set to 5mV/cm, time base=free run, Y1 mode, Y1 centered on the screen. Done all that prior to measuring)​

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TR833 - should be measured [-0.65V] at Collector​

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In the manual, TR833 appears to be a BC108 transistor​

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In the datasheet, it says that the collector is connected to the cap​

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So I measure ground (mount to the scope's chassis) to Transistor's cap....and get 18.83V (instead of [-0.65V])​

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Also, there's a 3rd question - should the checks, prior to the actual measurement, be done in this tedious way ? - looking for a datasheet of each transistor to identify the to-be-measured C/B/E ?

 

That is not the diagnostic approach I would follow. I would be looking at circuit functionality.
Your oscilloscope is about 80% functional.

The scope works with input on Y1 channel. There does not appear to be a trace corresponding with Y2 channel.

Look at the block diagram and circuit schematic and ask yourself what switch position would invoke Y2 display. With Trigger set to Free Run that would eliminate a trigger problem.

The Mode switch plays an essential role. I would get some switch contact cleaner or isopropyl alcohol and work the switch while applying the cleaning solution.

Next step would be to look at the Y2 amplifier circuit and see if the voltage is pushing the display off screen.

It helps to have an oscilloscope in order to diagnose this type of problem. Fortunately you do have a working oscilloscope. You can use the working Y1 channel to observe what is going on in the Y2 channel.

 

The Y2 amplifier is identical to the Y1 amplifier. so short both inputs. Switch them both to the least sensitive range.
Switch the mode to Y1 and adjust the position to be in the center of the screen. Starting at the input, write down the base and collector voltages of the transistors. Be careful to not short out anything when you take the measurements.
Switch the mode to Y2, center the position pot and measure the base and collector voltages. Where you get a different reading between the two channels will indicate the area of the problem in the Y2 amplifier.