Cruithne isn't really a satellite of the Earth. It's sometimes called a quasi-satellite, because from our perspective, it seems to make sort of a kidney-bean shaped orbit around us. But it doesn't really. It's orbiting the Sun, and although Earth's gravity does influence it somewhat, mostly the kidney-bean effect is just an optical illusion, much like epicycles (the little loops planets appear to make in the sky over time). There are several other quasi-satellites related to the Earth in gravitational resonances similar to Cruithne's. I couldn't find any information on Cruithne's composition, though.
Earth certainly can capture satellites, although the Moon's presence might make it difficult to keep them. It's a very large satellite for a relatively small planet, so it exerts considerable influence in the Earth-Moon system. A captured asteroid might orbit the Earth for a little while and then get flung out after an encounter with the Moon. Alternately, a captured asteroid might settle into an Earth-Moon lagrange point. L4 and L5 (co-orbital with the Moon) would be stable over long periods, and though searches haven't turned up any actual objects there, there does appear to be a greater concentration of interplanetary dust at those points.
Earth *did* capture a satellite recently, by the way. In 2002, an asteroid hunter name Bill Yeoung spotted an object that was moving surprisingly quickly, suggesting it was quite near the Earth. Before long, he'd concluded something startling -- it had to be orbiting the Earth. He reported his observations, the object was cataloged as J002E3, and other astronomers started looking at this new moon. It was a strange orbit, though, and one which could not have lasted very long -- calculations showed that it had to have either been placed into that orbit deliberately (a spacecraft, though it was rather large for such a thing) or captured naturally sometime in the previous few years. Nothing had been launched into an orbit like that recently, so it had to have been captured.
Its size was a bit of a concern. It appeared to be oblong, tumbling end-over-end (fairly common, actually -- the much larger 433 Eros rotates the same way), and big enough to cause serious damage -- it was very bright, which meant that it was either small and shiny or big and dull. So attention turned to its composition. Astronomers studied the object's spectra and found that it was actually shiny, which meant it was not very big. In fact, it was covered in titanium dioxide, a brilliantly white material. This gave them enough information to make a pretty good estimate of its actual size (which you can calculate if you know its reflectivity, its apparent brightness from Earth, and its distance from Earth, and if you make certain assumptions about its shape). But it was also puzzling; titanium dioxide is not very common in the solar system. But it is very commonly found in certain places on Earth -- paint factories. Titanium dioxide is a very popular white pigment.
That got the scientists thinking -- what if this is artificial? Meanwhile, people had been working on the object's orbit. Backtracking, they calculated that it had been orbiting the Earth once before. It had escaped from the Earth's gravity in 1971, the last time it had an opportunity to pass through a gravitational "keyhole" (one of the Lagrange points). And if you assumed the object was cylindrical and then ran the size calculations again, its brightness profile was perfectly consistent with an S-IVB, the third stage of a Saturn V rocket. The Apollo 12 mission had flown in 1969, but the S-IVB didn't have quite enough oomph to actually leave the Earth-Moon system. Instead, it flew past the Moon and settled into a highly elliptical Earth orbit. Astronomers lost track of it in 1970. Now it seems that they've found it again. ;-)
. . . for a while, anyway. After finding it again, they lost it again; the object has disappeared once more. Calculations expected it to depart the Earth system in 2003, so this is actually not surprising. It's far too small to be usefully tracked while it orbits the Sun. It's expected to come back again in 2032.
One upshot of working out its identity is that astronomers are no longer worried about it as an impact hazard. Given that it is hollow, it will not survive reentry. If it were rock, that would be another story, but it's not. Just a big chunk of space junk. ;-)
The S-IVBs of Apollos 8-11 are also out there, orbiting the Sun. They, too, might come back some day. ;-)
Here's an animation showing the capture of J2002E3.
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