embalmed, hermetically sealed inner casket which is then placed in an outer casket/container (e.g. airtray which is wood base fiberboard carton top, or wood crate)

yes, on same plane as regular pax. Yes, no notification to other pax (what purpose would that serve?)

Normally dead people are carried in caskets.On the same plane as passengers are going.People do not need to be notified as the casket is carried with luggage.

Your best bet is to contact the airline that you are flying with and they will give you specific details on what is required.