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.

Why is leise not inflected (no ending)?

Because it’s a predicative adjective after a linking verb (ist). Predicative adjectives stay in their base form: Die Heizung ist leise. You only add endings in attributive position before a noun: die leise Heizung.

What exactly does die Heizung refer to, and why the definite article?

Heizung can mean the heating system as a whole or, in everyday talk, the radiator in your room. The exact referent is given by context.
The definite article die is used because a specific, known heater/heating system is being discussed. Bare singular count nouns are rare in German in such contexts.

What does the semicolon do here? Could I use a comma or a period instead?

It separates two independent main clauses that are closely related.
In German, when you place two main clauses side by side without a conjunction, you normally separate them with a comma; a semicolon is also correct and signals a slightly stronger break; a period is fine too.
- Comma: … , hoffentlich bleibt das so.
- Semicolon: …; hoffentlich bleibt das so.
- Period: …. Hoffentlich bleibt das so.

What is hoffentlich, and how is it different from saying Ich hoffe, dass …?

hoffentlich is a sentence adverb that expresses the speaker’s attitude: hopefully.
Hoffentlich bleibt das so is concise and somewhat colloquial.
Ich hoffe, dass es so bleibt (or Ich hoffe, es bleibt so) is more explicit and slightly more formal. Meaning is the same; the difference is style and emphasis.

Why das and not es in hoffentlich bleibt das so?

das (demonstrative pronoun) points to the whole previously mentioned situation and is a bit more contrastive/emphatic.
es is also possible: Hoffentlich bleibt es so. It is slightly more neutral.
With so, das is very common because you’re deictically referring back to that state: that way.

Could I say Hoffentlich bleibt sie so since Heizung is feminine?

Yes. Hoffentlich bleibt sie so refers specifically to the heater (feminine), whereas das refers to the entire state of affairs. Both are idiomatic; choose based on what you want to highlight.

Where does so go in a clause like this? Can I say bleibt so das?

so is a predicative adverb meaning in that state/that way. It typically appears toward the end of the clause after the subject: Hoffentlich bleibt das so.
bleibt so das is ungrammatical. Acceptable variants include Das bleibt so and Das bleibt hoffentlich so.

Why is the present tense bleibt used for a future meaning?

German often uses the present tense for future reference when the context makes it clear: Hoffentlich bleibt das so naturally points to the future. You can also use future tense for emphasis: Hoffentlich wird das so bleiben, which sounds a bit heavier.

How would I negate the second clause?

Use nicht to negate the state: Hoffentlich bleibt das nicht so. Here nicht negates so (that way). If you want to negate something else, place nicht before the element you are negating.

Is the placement wieder leise fixed? Could I say leise wieder?

The natural order is wieder leise. leise wieder sounds wrong in standard German.
For emphasis you can front wieder: Wieder ist die Heizung leise, highlighting the recurrence.

Do I need a comma after Übrigens?

No comma is needed in Übrigens ist die Heizung … because übrigens functions as a normal sentence adverb within the clause. You might see Übrigens, … when Übrigens is used as a separate interjection introducing a new sentence, but that’s a different structure.

Why not say wieder leiser instead of wieder leise?

leiser is the comparative and means quieter (than before). wieder leise asserts that the heater is back to being quiet (a baseline state).
Use wieder leiser only if you mean it’s not fully quiet yet, just quieter than it was.

Would ist wieder leise geworden be better than ist wieder leise?

They are both correct but differ in nuance:

ist wieder leise states the current result state.
ist wieder leise geworden emphasizes the change of state (the process of becoming quiet again). Choose based on whether you want to highlight the result or the transition.