Learn Hardcore German: Die Kosten sinken, und dadurch sparen wir Geld. - The costs decrease, and because of that we save money.

dadurch literally means by/through that and points to the mechanism that produces the result. It links the second clause back to the preceding fact/event.
deshalb, darum, and daher mean therefore/that’s why and stress the result or conclusion.
In many everyday contexts they’re interchangeable:
- Die Kosten sinken; dadurch/deshalb sparen wir Geld.
  Nuance: dadurch highlights the means (the drop itself causes the saving), whereas deshalb/darum/daher simply states a consequence.

Why is there a comma before und? Is it required?

When two main clauses are connected by und, the comma is generally optional.
Both are acceptable:
- Die Kosten sinken und dadurch sparen wir Geld.
- Die Kosten sinken, und dadurch sparen wir Geld.
The comma can be used to signal a clearer pause or to make long sentences easier to read.

Could I just write two sentences instead?

Yes. Very natural and clear:

Die Kosten sinken. Dadurch sparen wir Geld.
You could also use Deshalb/Darum/Daher in the second sentence.

What’s the difference between sinken and senken? Can I also use fallen?

sinken = to decrease/decline (intransitive: the subject goes down by itself)
- Die Kosten sinken.
senken = to lower/reduce (transitive: someone lowers something)
- Wir senken die Kosten.
fallen is also intransitive and idiomatic with Kosten/Preise:
- Die Kosten fallen.
Other verbs: reduzieren, verringern (transitive) or sich verringern (refl./intransitive).

Why is die used with Kosten? Is Kosten singular or plural?

Kosten is plural-only (a pluralia tantum).
die here is the plural definite article: die Kosten.
Examples: Die Kosten sind hoch. Diese Kosten. Keine Kosten. There is no singular die Kost in this meaning.

Could I say Die Kosten werden gesenkt? How does that change the meaning?

Die Kosten werden gesenkt is a passive form meaning the costs are being lowered (by someone/through some action).
Die Kosten sinken just states they are going down, without implying an agent.

Where can I put dadurch in the second clause?

All of these are grammatical, with slight emphasis differences:

…, und dadurch sparen wir Geld. (fronted adverbial; mechanism emphasized)
…, und wir sparen dadurch Geld. (neutral)
…, und wir sparen Geld dadurch. (end focus; fine in speech, less common in very formal writing) You can also start a new sentence: Dadurch sparen wir Geld.

Why is it dadurch sparen wir and not dadurch wir sparen?

German main clauses follow the verb-second rule. If you start with an adverbial like dadurch, the finite verb must still be in second position:

Correct: Dadurch sparen wir Geld.
Incorrect: Dadurch wir sparen Geld.

Why is there no article before Geld? Could I say das Geld?

Geld is a mass/uncountable noun, so the default is no article: Wir sparen Geld.
Use das/unser/ihr Geld only when you mean specific, identifiable money:
- Wir sparen unser Geld für die Reise.
- Wir sparen das Geld, if a specific sum was mentioned before.

What’s the difference between sparen, ersparen, and speichern?

sparen = to save (money/resources) by economizing: Geld/Energie/Zeit sparen.
ersparen = to spare someone something or to save oneself something unpleasant:
- Das erspart uns Kosten.
- Ich möchte mir diesen Ärger ersparen.
  Not used for the general idea of saving money month by month.
speichern = to store (data); not used with money: Daten speichern. Don’t say Geld speichern.

Can I replace und dadurch with und damit or with wodurch?

und damit = and with that. It tends to refer to a concrete action just mentioned:
- Very natural: Wir senken die Kosten, und damit sparen wir Geld.
- Less ideal after Die Kosten sinken (no clear human action). It’s not wrong, but dadurch/deshalb generally fit better there.
wodurch requires a relative-clause structure:
- Die Kosten sinken, wodurch wir Geld sparen.
  This is grammatical and a bit more formal; it means by which we save money.

How would I say the same idea using a subordinate clause with weil or da?

Weil die Kosten sinken, sparen wir Geld.
Da die Kosten sinken, sparen wir Geld.
Both mean because the costs are decreasing. da is often a bit more formal/narrative, weil is the everyday causal conjunction.

Is starting a sentence with Und okay? For example: Und dadurch sparen wir Geld.

Yes, Und dadurch sparen wir Geld. is fine in speech and informal writing. Very formal prose sometimes avoids starting sentences with Und, but it’s not ungrammatical.

How do I negate this? Where do nicht and kein go?

To negate the first clause: Die Kosten sinken nicht, und deshalb sparen wir kein Geld.
- nicht negates the verb sinken.
- kein Geld negates the indefinite object Geld.
To negate only the second clause: Die Kosten sinken, aber wir sparen dadurch kein Geld.

How would I talk about a past event?

Use the perfect (common in speech) or preterite (in writing):

Die Kosten sind gesunken, und dadurch haben wir Geld gespart.
Die Kosten sanken, und dadurch sparten wir Geld.

What exactly is dadurch morphologically? Are there related forms I should know?

dadurch = da
- durch (a pronominal adverb: da- + preposition). It refers back to a clause/idea, not a concrete noun.
Related pairs:
- damit (with that), dafür (for that), darauf (on that), daran (on/at that), darüber (about/over that).
- Question/relative forms use wo-: womit, wofür, worauf, woran, worüber, wodurch.

Are there style/register alternatives to dadurch?

Yes. Neutral/formal: deshalb, daher, darum. More conversational: so.
Examples:

Die Kosten sinken, deshalb sparen wir Geld.
Die Kosten sinken, und so sparen wir Geld.

Does Die Kosten sinken describe intention or just observation? What if I want to say we plan to reduce costs?

Die Kosten sinken describes an observed trend/result.
To express intention/action:
- Wir senken die Kosten.
- Wir wollen/planen, die Kosten zu senken.
- Wir reduzieren die Kosten.

Pronunciation tips for the key words?

Kosten: KOHS-ten (initial k; clear o as in cost; final -en is a light schwa+N)
sinken: ZING-ken (initial s is voiced like English z)
sparen: SHPAH-ren (initial sp sounds like English sh + p)
dadurch: dah-DOOR-ikh (final -ch like the German ich-sound)
Geld: GELT (short e, final d pronounced like t in standard German)