Hello and welcome to HCM Judging by the symptoms and the time of appearance of the rash, it would commonly be a post viral or viral exanthem . That is rashes that appear due to a viral fever or flu. Also fever and increased sweating can cause rashes or heat boils esp on th back as the child sleeps. These rashes usually heal as the fever resolves. You may apply a calamine based lotion on the rashes. If it is itchy antihistamine syrup will help . I don t know if your pediatrician has prescribed one for the cold. Don t worry about vomiting out phlegm. They jus can t cough out yet. If the rashes change in appearance or don t get better do consult the doc again.