How close in style can the vocals/song be (the Vanilla Ice situation ~30 years ago):
That's up to the judge. In the case you cite, Vanilla Ice ultimately gave credit and a portion of the royalties to the original artists.
In today's litigious world, it behooves the DJ or whoever to offer the credit and split before it goes to court. Because they will be sued. Just as computers can mimic music, computers can also be used to detect a copy.