Hello I have gone through your complaints. What your doctor prescribed is good for your child for the treatment of throat infection
which can be due to bacterial infection a which can be confirmed by phlegm what you have noticed.
You can neglect the loose stools complaints unless it is distressing and leading to multiple episodes . If your child is maintaining adequate urine out put i.e at least passing the urine for 5-6 times then there is no need to worry.
If your child is having fever
then you can use calpol
is very early as you have suspected. Generally it will take a minimum of 6 months of age and there is hardly no correlation to loose stools and keeping the hands in mouth is a normal phenomenon.
Thank you . Good day