Is Rice's Theorem equivalent to the Halting problem?
Yes, you can prove Rice's theorem by reduction from the Halting problem.  See https://en.wikipedia.org/wiki/Rice%27s_theorem#Proof_by_reduction_from_the_halting_problem.
The reverse direction is also provable and is trivial.
In some sense all statements that true and provable are equivalent, in the sense that you can prove any by starting from any other, but that probably isn't a useful observation.