Before attempting to compensate a converter, you need to extract the control-to-output transfer function (TF) of the power stage: if a sinusoidal stimulus is applied to the control input of the said converter, how does the signal propagate through the converter to produce an observable response at the output. You have several ways to obtain this TF:
Analytical way: derive a small-signal model of the converter and graph its ac response in terms of magnitude and phase. This is a quite lengthy process which requires an equivalent linear circuit from a switching converter. My last book on the subject covers all the necessary steps for completing the process.
SPICE simulations: if you have an averaged model of the converter you study, then you can use it to extract the ac response of the power stage by first confirming the operating point is correct.
SIMPLIS simulations: SPICE can be used as a frequency-response analyzer (FRA) - see LTspice examples - but it is not very practical in my opinion. SIMPLIS does that natively and you can obtain a Bode plot of the power stage immediately from a switching circuit: no need to resort to an intermediate configuration, just use your cycle-by-cycle model. PSIM also offers this option but in a less easy way I believe.
Bench experiments: build a prototype on the bench and extract the response with a FRA.
It is important to understand what contributes poles and zeroes in a converter and going through 1. is important. This is because production, temperature and operating time affect the transfer function and your role, as a designer, is to neutralize its variability. Therefore, seeing where poles and zeroes are located in an equation is important. Then, you can use 2. or 3. to confirm the response and work on a compensation strategy. In the end, 4. will always be the referee, confirming or negating your hypothesis and calculations.
In your case, I will use one of the 60+ SIMPLIS free templates that I posted on my webpage which, for most of them, work with Elements, the free demo. The schematic of a current-mode (CM) push-pull is shown below:
From this circuit, we obtain an operating point (it is crucial to confirm the converter is properly regulating before considering the response - it is true for SIMPLIS but also SPICE of course) and the ac response of the converter:
Now, what is cool, is that I have automated the compensation elements from a macro in which you enter the desired crossover frequency and the phase margin you want. In this example, I have selected a 10-kHz crossover which is already a good value for a fast-responding converter featuring a 60° phase margin:
The components values around the op-amp are then available from the processed netlist.
Ok, so you now see the process - we've just scratched the surface here - for compensating a converter. Now, let's see your points in particular:
Do not think of a suitable crossover frequency by solely looking at the switching frequency. Crossover and phase margin selection depends on the transient response you need but also from a given robustness you may need at particular frequencies (i.e. input ripple rejection). Besides, crossover selection is often bounded by the converter itself when you have resonances like in a voltage-mode CCM boost or buck-boost converter for instance or if you have a right-half-plane zero (RHPZ) in the power stage TF. A RHPZ sets a limit for the crossover beyond which you'll face instabilities. So pulling out of thin air the crossover frequency based on the switching frequency alone is not a correct approach in my opinion. The more you push crossover, the more noise susceptible your converter becomes and you'll have to deal with issues linked to layout, noise pickup etc. As a preliminary summary, don't push crossover beyond what is really needed for your performance.
No, please go through the process I described as any converter is unique with its own parasitics and components values. With today's tools, the process is truly eased and you can do a lot on the computer before going to the bench.
Please see my answer in 1. and be reasonable with crossover selection.
When all is well stabilized, looking at the transient response is one way to check the converter adequately reacts when subjected to a load step: