Children of this age often suffer from a viral infection. The chest is generally clear in such conditions but may often be complicated by secondary bacterial infections. In apprehension for the same, your doctor had prescribed the antibiotics although it was not essential. Since you have started giving it, you should complete the full course of 5 days. But please understand that such viral infections take 5-7 days to recover irrespective of use of antibiotics. So you would have to be patient. T-minic as well as Tixylix are cough suppressants. I would not prefer them. You should rather give some expectorant like Relent syr. which will ease the situation by diluting the cough which needs to come out. Nasikind P is right to be given for blocked nose. If not relieved, you may use Nasivion Drops (after getting it prescribed by your doctor). Steam inhalation would be helpful for your baby. Just sit with the child in your lap and with the source of steam before you. The water vapour would be in the air and the child will automatically breathe it in. It will reach deep inside and help in diluting the cough. Please be patient as the condition might not respond to antibiotics but should still resolve in 5-7 days. In case you still feel apprehensive, get him rechecked by your doctor. The treatment is otherwise proceeding in right lines and you need not panic about your decision of going to this doctor.