Learn Farsi: شنیده می‌شود که اژدها هنوز زنده است.

شنیده می‌شود literally translates to "it is heard". It is the third-person singular passive form of شنیدن (to hear). Just like in English when we say "I hear that..." to gossip or share news, Farsi uses "it is heard that" as an impersonal reporting verb to mean "it is rumored that" or "word has it that".

Since there is a که (that) linking the two parts of the sentence, shouldn't the second verb be in the subjunctive?

Not in this case! While many verbs followed by که (like wishing, needing, or doubting) require the subjunctive mood for the second verb, reporting verbs like شنیده می‌شود (it is heard) or گفته می‌شود (it is said) are simply reporting a statement. Because they express a stated fact or belief rather than a wish or doubt, the following verb stays in the normal indicative mood (است).

The English translation puts "The dragon" at the beginning of the sentence. Can I do that in Farsi and say اژدها شنیده می‌شود که?

No, that would sound very unnatural in Farsi. English often makes the subject of the rumor the subject of the passive verb ("The dragon is rumored..."). Farsi strictly uses the impersonal "It is heard that..." (شنیده می‌شود که) followed by the actual statement (اژدها هنوز زنده است - the dragon is still alive).

Could شنیده می‌شود also be used literally, like "a noise is heard"?

Yes, absolutely. The verb itself is just the standard passive of شنیدن. If you say it literally means "a sound is heard". The "rumored" meaning only happens when it is used impersonally to introduce a whole new clause with .