Congratulations on your first scope. You should read the operation manual at least once over.Although it has schematics but it's not a service manual ( have tried to find but no luck) as it will tell which adjustments to make .
So you will have to trust your scope for now.
Unfortunately your friend is partly right but C131 is just a power supply decoupling capacitor.It may affect the circuit if bad but if C142 is bulging at the top of the can I suggest you replace it also.
First thing you should do is read the manual to set up the proper settings of switches of the scope. Look at your photo..Volt/Div Ch.1 is at 0.2V/Div if your test 1Khz. 0.5p-p (peak to peak) signal is accurate you should see waveform occupy 2.5 square vertical scale. I see it's about only 1 square = 0.2V p-p? Is the variable (center) knob on both Volt/Div of Ch.1 & 2 fully clockwise and locked/clicked? That is the reference position.
If at locked/clicked postion you should see peak to peak occupy 2.5 vertical squares.
Your signal in relation to vertical is on DC ( left side of photo ) is it same for Ch.2? Look aslo at the coupling switch for horizontal at the right hand side of the scope..is it at DC? Still ch.2 square wave should not be distorted? Please read page11 for probe high frequency adjustment at 10:1 switch setting at 50mV/Div. BTW try X1 switch setting on probe to see if ch.2 is same as ch.1? That's all for now.
Edit: If you have the chance try checking the 1khz. test signal if it's still at 1khz. Your calibration circuit has a 555 timer for frequency control with 2 pots and a 4011 logic for switching, mine only had a 4011 ic and no adjustment pots , only 5% resistors!. Only recently I tested it with my function generator with counter and found out it was at 895hz.
so it has aged..one of the resistors was double(original) the value of the schematic so I put in a 10 turn pot instead and replaced with 1% resistors. Now I have 1kz. according to my frequency counter.
So it was off since day 1 when I bought it brand new.
So I suspect many scope owners don't realise their cal signal has aged or drifted.