For your first question, would just refer to "something about a certain kingdom". Adding to get "" means "something that happened in a certain kingdom".
To give a similar example "" means "a trip via a bus". Removing the here would mean just "a trip of a bus".
So yes, your interpretation of this point is generally correct.
Since this looks like a fairy tale, if you were doing a proper translation you might use a more appropriate English phrase, like "Once upon a time in a kingdom".
For your second question, I think Choco's comment confirms the usage of the verb in question here.
I would translate this part more as "the coffin carrying the princess's body..."