My nearly 1 year old has had a runny nose and cough for a few days, and today developed a rash on his feet, legs, stomach, back, hands, face. He has no fever though. Could this be HFMD?

HFMD does not necessary comes with a fever first. You should bring your child to PD immediately to examine on the rashes to see if it is HFMD, as you need to quarantine your child at home instead of letting him to attend any class/school as HFMD is very contagious. This is a basic public responsibility.