Do I have to wire all inputs and outputs of the RTD-10 for it to operate correctly?

If you only need the relay output signals (R1: RUN or Defrost; R2: Fault), but don't want to use the hardwired inputs of the RTD-10 then that's fine. The indoor unit(s) can be controlled with the wired remote controller even if none or a subset of the inputs and outputs are wired.