Learn Hardcore Greek: Σήμερα δεν έχουμε απορρυπαντικό, οπότε δεν βάζω πλυντήριο. - Today we don’t have detergent, so I’m not doing laundry.

Greek negation for indicative verbs is normally δεν + verb (or μη(ν) + verb in other contexts).
Because you’re negating two separate clauses here (δεν έχουμε…, δεν βάζω…), you repeat δεν in each clause.

Why is it έχουμε (we have) but then δεν βάζω (I don’t put on a wash)? Who is the subject?

Greek often switches subjects naturally depending on meaning/context:

δεν έχουμε απορρυπαντικό = we (the household) don’t have detergent (shared situation)
δεν βάζω πλυντήριο = I’m not putting on a wash (the speaker’s action)

Subjects are frequently omitted because the verb ending already shows the person/number.

Could I also say δεν έχω απορρυπαντικό instead of δεν έχουμε απορρυπαντικό?

Yes.

δεν έχω απορρυπαντικό = I don’t have detergent (your personal supply)
δεν έχουμε απορρυπαντικό = we don’t have detergent (typical for a home/shared context)

Both are grammatical; the choice depends on what you want to imply.

What exactly does οπότε mean here, and how is it different from γιατί or άρα?

Here οπότε works like so / therefore / as a result introducing a consequence.

γιατί = because (introduces the reason)
άρα = therefore (more formal/logic-like)
οπότε = very common in speech for result, often similar to so in English

So: reason → result: Δεν έχουμε…, οπότε δεν….

Is οπότε always “so,” or can it mean something else?

It can also mean when/at which point, especially in relative/time uses. Example: I finished, οπότε έφυγα can feel like and then/at which point I left.
In your sentence it’s clearly the result meaning: so/therefore.

What does βάζω πλυντήριο literally mean, and is it an idiom?

Literally βάζω = I put, πλυντήριο = washing machine / wash.
As a set phrase, βάζω πλυντήριο means to put a wash on / start a load of laundry (i.e., run the washing machine). It’s a very natural everyday expression.

Does πλυντήριο mean the machine itself or the laundry cycle?

It can mean both depending on context:

το πλυντήριο = the washing machine (the appliance)
βάζω πλυντήριο = I start a wash / put on a load (the action/cycle)

In your sentence it’s the second meaning.

Why is the present tense used (δεν έχουμε, δεν βάζω) if this is about “today”?

Greek present tense commonly covers current situations and near-time decisions, especially with Σήμερα. It’s like English “We don’t have detergent today, so I’m not doing laundry.”
You could use future if you want to emphasize a future plan, but present is very natural here.

Could I replace Σήμερα with something else, and does its position matter?

Yes, you can use other time adverbs: αύριο (tomorrow), τώρα (now), αυτή την εβδομάδα (this week), etc.
Putting Σήμερα first highlights the time frame. You could also say Δεν έχουμε απορρυπαντικό σήμερα… which is also correct but slightly less “scene-setting.”

What is απορρυπαντικό exactly? Is it only detergent for clothes?

απορρυπαντικό is detergent/cleaning detergent, and in everyday context it often refers to laundry detergent.
If you need to be explicit, you can say απορρυπαντικό πλυντηρίου (= laundry detergent), but usually it’s understood from context.

How would I pronounce the tricky parts, especially απορρυπαντικό and οπότε?

Approximate pronunciations (stress in CAPS):

Σήμερα: SEE-meh-ra
απορρυπαντικό: a-po-ri-pa-ndi-KO (stress on -KO)
οπότε: o-PO-te (stress on PO)
πλυντήριο: pli(n)-TEE-rio (stress on TEE)
Greek stress marks (ά, έ, ί, etc.) show where the emphasis goes.

Why is there a comma before οπότε?

It separates two clauses:
1) Σήμερα δεν έχουμε απορρυπαντικό (reason/background)
2) οπότε δεν βάζω πλυντήριο (result)
In Greek writing, a comma before οπότε is common when it introduces a consequence clause like this.