You could... but what if you have a bimodal (or n-modal) distrbution?

If sequence does not matter and analyses are completely independent then it's simply a comparison of paired data. Randomization is a feature relating to the underlying process. If such process is meaningless, then there is no randomized label to attach.

Then it's simply a cross-over/before-after trial. But you could still randomize the sequence. For instance, half of the subjects will receive A and then B, and the other half B and then A.

I am not sure what you are talking about. Do you have multiple units of randomization per subject (eg two eyes in a man or four wheels in a car)? Then it's going to be a randomization with clustering. Quasi applies to non-random allocation, as far as I am aware (eg date of birth, which may be correlated with some features).