I am having a bit of trouble understanding how to optimise the Special Quasi-random Structure (SQS) for a given number of atoms beyond the default inputs.

Even in light of the relevant literature, I’m not sure which parameters (besides the cutoffs, which have been well explained) should be varied systematically to find the best SQS possible. For instance, what is the significance of varying:

n_steps: I presume that more steps is always better?

T_start, T_stop, optimality_weight and tol: Are there reasonable limits within which to work? I haven’t encountered any examples that do not use the defaults.

random_seed: Is it reasonable to expect the same output SQS for different random seeds?

I dont have much experience with using the SQS, but here are some thoughts.

The SQS is found through Monte-Carlo simulation, specially a simulated annealing simulation.
So more steps will probably lead to a better SQS.
Evaluating the artificial temperatures is probably best done by looking at acceptance ratios, I think ideally you want a very high acceptance ratio at T_start and a very low one or zero at T_stop.
But I guess you can also use some trial and error here and see what values produces the best SQS scores for your system.

The optimality_weight and tol are used to define the objective function for the simulated annealing, so they define what a “good” SQS structure looks like. Here I think you need to do some trial and error since they control e.g. how you weigh short-ranged order vs longer-ranged order. What is important for your system?
Maybe a good test would be to generate a few SQS with different optimality_weight and see how the property you are interested in changes with optimality_weight?

random_seed: Is it reasonable to expect the same output SQS for different random seeds?

I think its depends on your system, if you have a binary system and a supercell with 6 atoms then I think it quite likely that you will find the same SQS, whereas if you have a 5-component system and a 100 atom supercell then probably its not very likely.