Yes
https://wwwnc.cdc.gov/travel/notices/covid-1/coronavirus-india
Yes. Local health department of doctor should be able to provide the necessary vaccines. https://wwwnc.cdc.gov/travel/destinations/traveler/none/india
Yes. It can be obtained online through Indian Embassy Website. More information - https://www.indianembassyusa.gov.in/extra?id=87
Madurai City, Tamil Nadu, India
No, there is no direct flight. You would have to take a domestic flight from Chennai to Madurai.
Bring cash Dress in cotton or Linen. Just have a rainjacket in case it rains. Drink bottled water Expect traffic (atleast in Chennai) Some local places might not have Western toilet, be prepared. Wedding ceremony in India is in a different format than the western ones. You could watch South Indian wedding youtube videos to have an idea. I am sure everyone will have a great experience. Hope you enjoy visiting India :)