I guess the example below provides one answer to your first question.
A famous positive-definite function** is the one in the Bessis-Moussa-Villani conjecture:
Let $A$ and $B$ be $n \times n$ Hermitian matrices. Then the function $$\phi(t) = \mbox{trace}(e^{A+i t B}),$$ is a positive-definite function.
** Conjectured to be positive-definite, though apparently it has been proved very recently; however, until that has been verified independently, I will adhere to the safety of the word "conjecture"
