Thanq for your query.
.First antibiotic is not required as your child is not having fever and as your child is having cough cough syrup with mucolytic
is due to antibiotic associated and start one oral proboitic once or twice daily and you can give oral rehydrating solutions to prevent dehydration
due to diarrhea.
.Dont start any antibiotic without doctor consultation and prescription as it may be inappropriate for your child or not required by your child and indiscriminate use will cause drug resistance
Dont be panic and consult your Pediatrician
for the examination of your child and for proper medication.