"A law that applies in every situation" is an instance of empirical universality, which is the weaker species of universality. Kant's entire problem with this is that a principle that has to be applied to a case in order to be true can admit exceptions since the only way to actually prove it true is to intractably observe every situation, so instead the warrant for the principle ends up being a mere inference. On this point he's specifically going after Hume: If I can only infer that the sun will rise tomorrow because it rose yesterday and the day before, I still leave open the possibility that the sun simply won't rise tomorrow. This is how Hume started questioning necessary causality and Kant totally freaked out.

Pure Universality is strictu sensu synonymous with necessity because a principle of this sort is a condition of possibility. This example is technically incorrect, but for illustrative purposes all stones either necessarily fall back down to the ground when thrown up in the air or I'm not able to write this. Kant thought causality in much the same way, either human beings can necessarily conceive of it without any empirical support or experience in toto is impossible.

Kant explicitly wants to derive morality from Pure rather than empirical reason. Self-interest, altruism, pleasure, and even our everyday notion of happiness are set by the wayside in the name of actions that wholly brought about by principle than any expectation of reward, service to a higher cause, or just plain conformity (which are all arguably the same thing).

The key thing to realize is that Kantian morality is really really inhuman: for all intents and purposes, Kant wants to automate morality. Hence my language of "Act as if you were a force of necessity".