First, I'm not sure "tear" is a good term for it. Second, space-time is FOUR dimensional (in Special Relativity, at least), not three. A better way to think of it is that all masses cause a localized curvature of spacetime; the bigger the mass, the greater the curvature. A Black Hole forms when the mass is SO large (and the density so high) that the curvature of Space-time becomes infinite.
All depictions I've ever seen of Black Holes show them to be like infintely deep whirlpools. I assume that these holes "fall off" in all directions, depending on their formation.
So what determines the direction in which a Black Hole forms? Are there variations in the "density" of space-time, such that different Black Holes "fall off" in different directions?
There's no "direction" the Black Hole is falling "into." The Black hole is located exactly where the center of mass of the material that formed the Black Hole was just BEFORE the BH formed. Another way to think of it is that the localized Spacetime curvature caused by a mass has no "direction" either; it causes a uniform curvature in all directions. The space-time curvature of the BH is only moreso.
Also, do all known black holes "rotate" in the same direction?
The short answer is I'm not sure, but I expect the rotational distribution of BH is about the same as the rotational distribution of other celestial matter, such as stars, clusters, galaxies, etc. It is interesting that you ask, in that rotation (actually angular momentum) is one of the few physical characteristics a BH has. The others are mass and (I think) electric charge.
I hereby defer to those more knowledgeable than I am on this subject whom I've pinged to revise and extend my remarks as needed to correct any errors or ommissions.
What if spacetime curvature and mass are coincidental, circumstantial, and are not cause and effect. Or what if spacetime curvature causes mass? There was a comment quite a while back pertaining to the old vortex theory of gravity, that if a planet were not there, say, Jupiter, its gravitational field would still be present, that the presence of gravity causes the planet to form right there. This is just to point out that these are all hypotheses and that flat statements might close some doors to speculation.
I don't think electric charge survives. Just mass and spin. But the mass which was the big bang probably wasn't even spinning, in that there was no reference frame for determining spin.