What is the form of zeta function of an elliptic curve over $\mathbb{F}_p(t)$? Does it satisfy a Riemann hypothesis?

It is a polynomial in $p^{-s}$ and yes, it satisfies the Riemann hypothesis. See for example Theorem 2.2.1 of Lecture 4 of Douglas Ulmer's *Elliptic curves over function fields* available on the arXiv: 1101.1939v1.