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?