Have Cat Questions? Ask a Cat Vet Online.
Cats can definitely have colds, otherwise known as upper respiratory infections. URI's will come and go and are mostly harmless. Keep an eye on the secretions from the eyes and nose. If it turns yellow or green, you need to see the vet. Otherwise, you can let it run its course. Be sure that the cat is eating and drinking because they can quickly go downhill if they get dehydrated.
The cat could have brought in fleas AND you should also be worried about FIV/FelV.
The new cat needs a vet visit right away.
If you see any green or yellow mucus from your cats, you need to see the vet with them because that means that an infection may be present.
Let me know if you have more questions.