Translation:The king is dead. Long live the king!
The word "да" in fact has more meanings then just "yes."
In this case it is a particle that most accurately can be translated as "let" or "may". "Да здравствует король" literally means "May the king prosper (or be healthy)"
-Да будет свет! - Let there be light!
-Да не обеднеет ваш дом! - May your house never grow poor!
When you say hello to people in Russian are you saying long live you?
"Здравствуйте", or "здравствует" does not literally mean "long live", but rather "prosper" or "be healthy", as I mentioned above, but in this sentence "long live" is an idiomatic translation.
Russians don't use the word "царь" to refer to European monarchs and this particular phrase was not used in regard to Russian monarchs. Thus using "царь" or "император" here is pretty much nonsensical. This is a traditional translation of an idiomatic expression. These things tend to have a fixed phrasing.