Does the "common cold"/flu cross species barriers?

I’m sick. Is it ok to spend time around animals (specifically cats and dogs), or would I risk transmitting something to them, especially if I sneeze on my hands and they smell/lick 'em afterward?

I don’t know the answer to your specific question. I imagine it would have to do with what partcular strain of the flu you have.

But, yes, the flu can cross species barriers. It’s actually one of the main reason why we haven’t already eradicated the flu. The virus lives in more populations the just humans. fowl and pigs are the two other main ones I believe. What happens is every year the virus jumps from one species to another and a recombination of the Species 1 flu recombines with the species 2 flu to create a new and brand new flu that has characteristics of both. lather rinse repeat and you have a new slightly different flu strain every year.

The common cold and influenza are two completely unrelated diseases. You might just as well refer to the “common cold”/herpes or the “common cold”/smallpox.
Influenza and colds can’t infect dogs and cats. Pigs and birds certainly get influenza, but not dogs and cats. I’m not aware of any animal host for the human cold virus.

This question was recently discussed here. To repeat my contribution, ferrets can catch human colds, so try not to sneeze on the poor dears.