As for Throbert's No.(3) if it's cuisine, it's always в - it's an abstract noun. В итальянской кухне, в кухне Италии.... и т.п.

На vs в - if it's на кухне it means that something/someone is in the kitchen or pertains to the kitchen,
while в кухне - implies the physical area, room (kitchen) when it's necessary to distinguish it from other areas. So на кухне is used most of the time.

A building contractor (eg building a new home) may say "В кухне надо сделать ещё один проём под окно" - Another window must be provided in the kitchen. Here в suits better as it implies the physical area of the home being built (i.e. kitchen area).