$$ x x $=$ Rumpelstiltskin is a valid $($necessarily true$)$ sentence of first$-$order logic.

False. This sentence is not valid in first$-$order logic.

True. This sentence is not valid in first$-$order logic.

False. This sentence is not satisfiable because there are no objects that satisfy the equality predicate.

True. Any existentially quantified sentence in first$-$order logic is true in any model that contains exactly one object because there is always one object that satisfies the quantifier.