As I understand it, a Black Hole is a "tear" caused in three-dimensional space-time by an infinitely dense object.
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?
Also, do all known black holes "rotate" in the same direction?
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.
The whirlpools that are drawn around black holes (as in the above illustration) are accretion disks. Accretion disks form because the infalling matter will, in all likelihood, have a nonzero net angular momentum. The angular momenta of the infalling objects all cancel each other in every direction but one (the axis of the disk). The matter left in the disk goes into orbit around the black hole.
The accretion disk of a black hole actually wouldn't look like a normal accretion disk as shown above. This is because light bends around the black hole, with the result that if you are looking at the near side of the disk almost edge-on, you'll be looking at the far side of the disk face-on. It looks sort of like a misshapen LP record with a 90-degree fold in it. (See an example here.)