Unless the pots had issues, I would have left them alone. Depending on what you put in, and where, some 10-turn pots are wirewound and the internal coil of wire adds inductance to the circuit. In an amplifier that runs up to 50MHz, it could be significant. I really have to emphasize "if aint't broke, don't fix it". Especially in the middle of a repair where the thing you're replacing is known working. It adds more unknowns to what might be wrong.
Transition time (aka rise time or fall time), is how long a signal takes to get from one plateau to another. A square wave is used for the calculation here. A change from one voltage level to another doesn't happen instantly, and can be seen as a fast ramp. The measurement for transition time is the points on the ramp between 10% and 90% of the starting and ending voltage levels, where the difference between starting and ending voltage levels is defined as 100%. I don't have a Rigol scope, but I'm sure it has a button to measure it automatically for you, probably under the "Measure" menu. You need to have your horizontal sweep set fast enough so that you can see the ramp of the signal. If you're curious what it's doing, search for "measure rise time on an oscilloscope" for the old way of manually doing it.
The square wave is not looking great, and it may be related to the crossover issue. What does it look like at a lower frequency, say 10kHz? Looks like you have it set to 100kHz in the screen shot.
For the unusually high output voltage, I would start with verifying the amplifier input at TP4 as described on page 10.4-1 1. If the input isn't right, certainly the output won't be.
In the one scope capture you say the purple is "tracking output"? Do you mean the trigger output?
The crossover distortion could be caused by very mismatched hFE pairs, or it could be something wrong with the bias in one of the stages. The transition from the positive side of the amplifier driving to the negative side, and visa-versa, is discontinuous. You could try changing the frequency of the waveform and observe what it does to the distortion. If it's less of a problem at lower frequencies, and the distortion gets closer to 0V, it's a problem with the speed of the changeover, perhaps capacitive in nature.
Also, note that the adjustment procedure has you constantly putting in attenuators and taking them out and changing the multiplier on the scope. This is because HP has chosen an insanely over-powered 20GHz scope for the adjustment procedure that can only do 1mV/div to 80mV/div. So, all these attenuators are compensating for the scope's deficiencies in this application. Instead, you can simply put a 50 ohm terminator on your scope (or use the internal 50 ohm termination if it has one) and set the scope according to the needed V/div for any particular step.