It seems odd to me that no other asteroid around the Earth is even close to the size of the Moon. Anyone have any idea about the size and orbit radius of Earth's "second moon" ?
There are no other known natural satellites orbiting the earth. Most of the objects orbiting the earth are man-made and largest object other than the Moon orbiting the earth is the International Space Station which orbits the earth at a height of about 400km from the surface and is about 100m wide.
It may be odd to you that there are no other natural satellites orbiting the Earth. We are not sure but it maybe that the probability of an object ending up in a stable orbit around the Earth is not that high. There are roughly 3 different categories of natural satellites - Large moons in roughly circular orbit like the Moon, small rocks in low orbits formed as a result of collision and nearby asteroids that a planet catches in its gravitational well.
It maybe that there are Earth is not massive enough to capture fast moving asteroids in orbit. Or the Moon which is quite large relative to the earth destabilizes a satellite's orbit around the Earth and there are not many stable orbits that a random natural object can get itself in.
First of all any body revolving around a planet is known as its satellite, not as an "asteroid". And at this time Earth only has one natural satellite which is the Moon (but there are few theories suggesting that Earth once had two moons but they merged into each other due to gravitational pull between them and formed a single moon).
Asteroids are the masses of rock, gases, minerals etc. which are generally revolving around the star (in our case it's the Sun). Even in our solar system we have an asteroid belt between Mars and Jupiter in which thousands of asteroids are orbiting around our star. But asteroids don't have much mass and are small in size so they are not characterized as planets, though sometimes larger asteroids are known as planetoids.
By the way the largest asteroid yet found is named Ceres. It is about 1/4 of the size of our Moon and it is also orbiting around the Sun like other asteroids.
Temporary's have come and gone.
To the best of my knowledge there are no natural satellites of Earth other than the moon. If you know of one, it has only just been discovered, in which case it should feature on the TV news tonight, but I think more likely you are confused by something you've read in science fiction. Any newly discovered second moon would have to be very small, very far away and have a low albedo to have escaped detection for all these years.