Flu vaccine is recommended for everyone from 6 months and above.. So YES, you can give your baby.

The crying may not be because of the flu. This is what you need to do :Just keep warm, keep breastfeeding. If it persists, gives fever and leads to difficulty in breathing, please seek medical help immediately by taking the child to a Paediatrician.

Snoring in babies could be positioning during sleep, habit especially if familiar or adenoids enlargement. Kindly see a Paediatrician for proper evaluation and management

You need to see a Paeditrician first for a proper evaluation to know why he is having recurrent illnesses or infections. That will determine the best thing to do in terms of advice

Do steam inhalation for him with some drops of eucalyptus oil aka silver bird. Also give him piriton syrup 2.5 mls at night only for 3 daysm

Are you sure it's an antibiotic drug?