There seem to be a number of versions of this scope and it's hard to say whether the manual I've seen corresponds exactly with yours.
From the circuit diagram you've shown the square wave generator looks like the calibrator and consists of an astable (T57 and T59) with the output through an emitter follower (T59). The circuit looks quite simple. The purpose of this is to present a square wave at around 1kHz and an amplitude of somewhere around 0.5V ( in this case 0.2V) to compensate a 10X scope probe properly. You connect the probe to the calibrator output and adjust a preset capacitor on the probe to display a square wave without overshoot or undershoot. In other words a square wave with a 90 degree angle, which looks like a square wave from a text book.
If you use a probe on its 1X setting or use something like a BNC lead with crocodile clips, you ought to see a square wave.
Usually with basic analogue scopes like this, you can adjust the amplitude of the square wave by adjusting a preset potentiometer, but the frequency is fixed and is only approximately 1kHz.
If you don't have another square wave source, you can make one up using a 555 timer or a relaxation oscillator using TTL. Those ebay function generators using an 8038 or XR2206, are rather limited but look good for their low price.
Unfortunately with oscilloscopes you soon get to a stage where you need another oscilloscope to fix them easily.
The sawtooth looks as if it's supposed to be 5V peak to peak. I can't see it on the circuit diagram I've seen. I can't imagine it's very useful and if the rest of the scope works, I wouldn't worry about it at this stage.
As for the voltages on the output of the X amplifier, if you can centre the display properly, don't worry about them. That's to say, when it displays a line with the trigger on auto, you ought to be able to use the X-shift control to move the line to the right, so the left end is visible and to the left so the right end is visible. There's usually an adjustment so that when the X-shift is in the centre position, the centre of the line matches the centre Y line, but if that isn't quite right it isn't normally a problem.