Learn Hardcore German: Übrigens ist die Heizung wieder leise; hoffentlich bleibt das so. - By the way, the heating is quiet again; hopefully it stays that way.

Because of German verb-second (V2) word order. Übrigens sits in the first position of the clause, so the finite verb ist must come second, and the subject die Heizung follows it. Without a fronted element, you would get the more neutral Die Heizung ist wieder leise.

What does Übrigens add, and where else can I put it?

Übrigens means by the way; it marks an aside or incidental remark, not an additional argument like außerdem.
Typical placements:
- Sentence-initial: Übrigens ist die Heizung wieder leise (slightly emphasizes the aside).
- After the subject: Die Heizung ist übrigens wieder leise (very common, more neutral).
End position is uncommon here; stick to the two positions above for natural style.

Why is wieder used here and not nochmal, erneut, or schon wieder?

wieder = again, back to a previous state; perfect with stative predicates: wieder leise.
nochmal/noch einmal = one more time (repetition of an event), not idiomatic with an adjective like leise.
erneut = again but more formal/literary; with adjectives it can sound stiff.
schon wieder = yet again/already again, often with a nuance of annoyance.