Codeine is NOT a morphine prodrug. More than 80% of codeine is metabolized into codeine-6-glucuronide (C6G) and that is the metabolite responsible for codeine's side effects. Only 3-5% of codeine is O-demethylated to morphine, which is quickly excreted anyways. The rest is is N-demethylized to norcodeine.
Heroin is a morphine prodrug - 100% of it is converted into morphine. Yes, even through the IV route, first it's converted into morphine and a very small percent is converted into 6-MAM, which is also converted into morphine eventually.