Can pets catch the cold?
Yes they can. The viruses that cause a cold include rhinoviruses (up to 80%), coronaviruses and influenza (together accounting for much of the remaining 20%) and a small percentage of adenoviruses. When discussing cross species transfer, influenza is the most important to discuss as it is a fairly common cause of the "common cold". 

Influenza A has humans, pigs, bird and horses as its host. Different influenza strains have different hosts. Influenza can recombine if two strains infect the same host (antigenic shift) and also mutate (antigenic drift). With a combination of these influenza can and has numerous times crossed the species barrier which has caused the major flu pandemics as well as swine and avian flu. It's this reservoir that makes these viruses so hard to eradicate. When we are the only host such as in polio virus, it is a lot easier. Pigs don't adhere to national flu guidelines. 

Comparing recovery times, all mammals are roughly similar although due to our slightly more evolved immune system as well as social factors of knowing what to do when we are ill we recovery very slightly earlier.