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Please do not panic. I am a medically trained Siddha doctor and I can assure you that it is a common practice in many Indian households to give the herbal home remedies--like kadukkai, massikkai, thanrikkai and vasambu after the baby is 1 month old. They rub the above mentioned herbs and mix with breast milk to make a paste and put it in the tongue of the baby to improve the digestion, immunity and to prevent any flattulence in alternative days or after giving oil bath to the baby. It is one of the 3 herbs in Triphala/Thiribala, the other 2 being nellikai (Amlaki ) and Thanrikkai (Biphitaki).
It is a best astringent and also a tonic to the body. It is one of the regular ingredients in siddha/Ayurvedic drug remedies given for dry cough and cold like Thalisaathi chooranam/chooran, Thribala chooran etc.
You also can give one more drop of the kadukkai...if needed before vaccination.
However, being a mother, if I were you, I would keep a close eye on the babysitter and put my foot down to make sure that she doesnt give anything to the baby (even out of goodwill) without my knowledge, in future.
2. To your second question, no they do not vaccinate the baby when she has fever/severe cough. But the throat noises you mention need not to be phlegm. It may be due to the feeding posture maintained and the baby gurgling the milk. If you bring her to the paediatrician, he will use his stethoscope to see if her lungs are clear and then will decide whether to proceed with the vaccination.
Hope my answer helps you. Happy Parenting!