Energy drain is a supernatural ability that specifies that it requires an attack to apply, but existing examples (e.g. the wight) attach it to other, existing attacks, rather than making it a separate attack unto itself.
For the soul eater, then, it appears to be attached to a “touch” but not a separate touch attack. Energy drain still requires an attack roll, so
strikes a foe with a natural weapon
Yes. This is an attack that involves physically touching the target.
strikes a foe with [...an] unarmed strike
Yes. This is an attack that involves physically touching the target.
casually brushes up against someone in a crowded marketplace
No, no attack roll is used here.
Not mentioned, but touches a foe with a touch-attack effect
Yes, this would also be an attack that involves physically touching the target, so the soul eater would drain energy on top of the usual effect of that attack.
Also not mentioned, but touches a foe with an attack just to drain energy
I would say yes, in that I believe a character can choose to roll a touch attack at any time to attempt to touch a character, and for a soul eater that touch would drain energy. However, to the best of my knowledge, there is no rule that you can choose to use an attack to try to touch someone outside of a specific touch-attack ability (which the soul eater class does not grant).
Overall, I do not like the soul eater class. It is very much a one-trick pony, and it’s an extremely binary trick: either it works and the target is pretty much dead, or it doesn’t work and the soul eater is useless. That means as a DM you have to either use immune creatures against this PC all the time, and that character will struggle mightily to contribute since half their levels are devoted to this energy drain thing, or else allow this character to completely dominate combat encounters.
Personally, I would not be wild about having a soul eater in my party, and would want to have a long chat with the player about how to produce their character in a manner appropriate to the game, either by finding a replacement for soul eater or else tweaking/rewriting the class to both be less devastating with the energy drain and also have other things going on.