Translation:The king is dead. Long live the king!
In Hebrew it's actually traslated "The king is dead, long live the new king" to make the meaning clearer
does да not function as yes here or is it a part of long live? When you say hello to people in Russian are you saying long live you?
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.
The title король was never used by any Russian sovereign but rather for western rulers, like the Swedish or Polish king. Why is this word here rather than царь or император which were titles actually used by the Russians?
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.