sheik most definitely beats ICs. she can outcamp them, should never get grabbed, one combo means nana dies if you're diligent, and it's very easy to destroy popo 1v1.
i really don't think it's worth discussing the differences between fox/marth/sheik, it's like a 9.9 to 9.8 to 9.7 -- it's not a big enough difference to really matter for the vast majority of people. it's just for people playing theory craft.
but theory craft is fun so i'll bite.
falco counters sheik if falco camps. a smart campy falco can just dissect sheik apart and it's really hard for sheik to ever mount anything against it. falco's margin of skill to pull it off is much higher than sheik's, which is why i think a lot of people call it even or say sheik has the advantage or w/e, but once you get to that skill with falco you definitely have the advantage.
i think sheik v fox is pretty even, it's just that the margin of skill has to be higher for the sheik player. sheik just can't mess up, but if she doesn't -- which at the pro level she won't -- then it becomes brutal for both sides and personally one of the most frustratingly fun matchups in the game.
i think marth's severe problems with both falcon and sheik make him unable to contend with fox and sheik for the crown. but if you believe ICs actually counters sheik then i see no way you can have sheik as above or equal to fox, who is really on a plane of his own.