Learn Hardcore Indonesian: Dokter bilang bahwa demamnya ringan, tetapi batuknya perlu obat. - The doctor says that the fever is mild, but the cough needs medicine.

-nya can mean either third-person possession (his/her/its) or “the/that” (a definite item already known in context). Here it most likely refers to the patient already being discussed.

demamnya ringan = his/her fever is mild / the fever is mild (context decides)
To be explicit: demam dia ringan (okay but less natural than demamnya), or demam anak saya ringan (“my child’s fever is mild”)
For a definite “the”: demam itu ringan (“that/the fever is mild”)

Do I need bahwa after bilang? Can I drop it or use kalau?

You can drop bahwa after bilang, and in speech many people use kalau instead.

Informal: Dokter bilang demamnya ringan…
Very common in speech: Dokter bilang kalau demamnya ringan…
More formal: Dokter berkata/mengatakan bahwa demamnya ringan…

What’s the difference between bilang, berkata, mengatakan, and katanya?

bilang: informal “say/tell.” Common in conversation. Example: Dokter bilang demamnya ringan.
berkata: more formal “said/spoke.” Often with bahwa or a direct quote. Example: Dokter berkata bahwa…
mengatakan: formal “stated/said.” Transitive; takes an object clause. Example: Dokter mengatakan bahwa…
katanya: “(they) say/said” or “according to him/her,” often used for hearsay. Example: Katanya demamnya ringan.

What’s the nuance difference among tetapi, tapi, namun, and sedangkan?

tetapi = “but” (more formal/written). Often preceded by a comma between two clauses.
tapi = “but” (informal/colloquial). Very common in speech.
namun = “however.” Usually starts a new sentence or clause: Namun, …
sedangkan = “whereas/while” (contrasts two different subjects/actions): Demamnya ringan, sedangkan batuknya perlu obat.

Why is ringan placed after demamnya?

In Indonesian, adjectives typically follow nouns. Examples:

batuknya parah = the cough is severe
demam tinggi = high fever
dokter muda = a young doctor

Can I say demam ringan instead of demamnya ringan?

They mean different things grammatically:

demam ringan = “mild fever” (a noun phrase). Example: Dia kena demam ringan.
demamnya ringan = “the fever is mild” (a full clause). Example: Dokter bilang demamnya ringan.

Is batuknya perlu obat natural? Or should it be perlu diobati or perlu minum obat?

Batuknya perlu obat is natural and common. Alternatives:

Batuknya perlu diobati (more clinical/formal: “the cough needs to be treated”)
Dia perlu minum obat untuk batuknya (focuses on the person taking medicine)
Dia perlu obat batuk (she/he needs cough medicine) Note: Batuknya membutuhkan obat is grammatical but sounds stiffer/less idiomatic in everyday talk.

Is batuk a noun or a verb here?

Here it’s a noun (marked by -nya, and it’s the thing that “needs medicine”). Compare:

Verb: Dia batuk (“He/She is coughing”), Dia batuk-batuk (coughing repeatedly)
Noun: Batuknya parah (“His/Her cough is severe”), Obat batuk (“cough medicine”) You may also see berbatuk (“to have a cough”), though plain batuk is more common.

How do I say “the doctor” vs “a doctor” in Indonesian?

Bare dokter can mean “the doctor” or “a doctor,” depending on context.
dokter itu = that specific/the particular doctor.
dokternya = the known/previously mentioned doctor, or “his/her doctor.”
seorang dokter = a doctor (introducing an indefinite person). Example: Seorang dokter bilang… (“A doctor said…”) vs Dokter itu bilang… (“That/the doctor said…”).

Is the comma before tetapi correct?

Yes. When tetapi connects two independent clauses, a comma is standard: …, tetapi …. With the informal tapi, many speakers omit the comma in short sentences, but it’s fine to keep it.

Is the register mixed here? (bilang is informal, tetapi is formal.)

Yes, it’s a mild register mix, but natives do this often. To keep consistency:

Informal: Dokter bilang demamnya ringan, tapi batuknya perlu obat.
More formal: Dokter berkata/mengatakan bahwa demamnya ringan, tetapi batuknya perlu diobati.
Neutral alternative: Menurut dokter, demamnya ringan, tetapi batuknya perlu obat.

How can I make it crystal clear whose fever/cough it is?

Add an explicit possessor or name:

Demam saya ringan, tetapi batuk saya perlu obat.
Demam anak saya ringan, tetapi batuknya perlu obat. (the second -nya refers back to “anak saya”)
Demam pasien itu ringan, tetapi batuk pasien itu perlu obat. Context is usually enough, but these forms remove ambiguity.