I need to ask few more questions before answering your query. Early consultations can prevent complications. Kindly contact me on whatsapp chat  (84960595 six five) Dr Faisal Zoheb MBBS , MD PAEDIATRICS , PGPN (BOSTON), Neonatology fellow ( Newborn and vaccination specialist )

Yes give 5ml morning afternoon night for 3days

Better to give home remedies like "kada" made with Tulsi, honey, ginger. Or nebulisation with levosalbutamol. Better to avoid cough syrups

You can give safe OTC options: honey (1 tsp 2–3 times/day), warm fluids, steam inhalation, and paracetamol if fever. Vomiting can be due to severe coughing. Avoid multiple syrups together. Other medicines (like bronchodilators or anti-allergic) should be taken only after pediatrician advice. Consult doctor if cough persists >5 days, breathing difficulty, or repeated vomiting.