Bubba beat me to it
Just wanted to add that zoonotic diseases are relatively rare. The Hendra virus is particularly nasty and associated with a high mortality but thankfully only ever been reported sporadically in Australia. One outbreak was fairly recent too, this year I think.
As for cold/flu viruses cross infecting species, well, it is possible but again rare as a fairly specific recombination event (change in DNA/RNA of virus) is necessary. The result can be a virulent pathogen that often spreads rapidly - remember swine flu?
Can happen but unlikely.
Bacteria are more prone to cross species infections. E. Coli will inhabit just about anything that is around 37 degrees C.