The first animal in orbit was Laika, a Soviet space dog. She was launched aboard Sputnik 2 on November 3, 1957. The first animals in space were fruit flies, launched by the United States aboard a V-2 rocket on February 20, 1947.

I wonder whether some large asteroid hit the earth and blasted some poor animal into space some millions of years ago...

If an animal were imparted enough energy to be thrown into space, it would not get there intact.

Tiny spiders can float high enough in the air that it might be possible for some to be carried beyond the Karman line.