Yes, US citizens are required to have a visa to enter India. You can apply for an e-visa here: https://indianvisaonline.gov.in/evisa/tvoa.html
https://wwwnc.cdc.gov/travel/destinations/traveler/none/india The above link from the CDC includes a thorough list of recommended vaccines and medications, as well as guidance for staying healthy while traveling in India.
In Kochi and the surrounding areas, the temperature will typically be in the upper 80s, with high humidity and rain in the late evenings. Bring some warmer clothing if traveling to the hill stations of Kerala. Temperatures may drop to the 50s in the evening, and rain is a frequent occurrence.
Yes! The city is also known as Ernakulam.