Brief Answer:
Continue the same and observe the baby

Detailed Answer:
Hello

Thanks for asking on this platform

I can understand your concern.Cough and cold in this age without any history of fever is most commonly due to some allergy/hyperesponsiveness of airways.
It may take 7-10 days sometimes even more to get complete relief from cough with the medication.
CZ3 contains cetrizine which is an antiallergic,ventorlin contains salbutamol which is a bronchodilator(relieves airway spasm) and levalbuterol is also a bronchodilator which provides short term relief.
I won't get worried if cough has not worsened,not associated with rapid respiratory rate,excessive irritability or significant poor feeding.
As steroid helps in decreasing the inflammation and counteracts the hyperresponsiveness,short duration of prednisolone(steroid) in such cases can be added after consulting your pediatrician for better relief.

I hope this helps.
In case you notice any breathing difficulty,excessive irritability consult your physician.
You can ask me further if you have any more doubts.
Wishing for your child's speedy recovery.
Regards

---

Brief Answer:
There is no relation between vomiting and mucus

Detailed Answer:
Thanks for follow up

It a common myth that vomiting provides relief from cough as mucus comes out in vomitus. Mucus is produced in the lungs(in common terms in the chest).Vomiting with cough is due to the irritation in the throat and it is the swallowed mucus which comes in the vomitus from stomach not from the chest(or lung).
You can give combination of mucolytic like ambroxol(mucus thinning agent) with salbutamol(like ambrodil s)instead of ventorlin.

Thanks and Regards

---