Learn Hardcore Finnish: Älä kaada kahvia matolle, koska se on vaikea puhdistaa. - Don't spill coffee on the rug, because it is hard to clean.

You’re talking about an indefinite amount of coffee (not “all the coffee” but “some coffee / any coffee”).
- Älä kaada kahvia ≈ “Don’t pour (any) coffee…”
- Älä kaada kahvi would sound wrong; with substances and mass nouns, Finnish almost always uses partitive when the amount isn’t specified.
With many verbs, especially when the object is not a complete, bounded whole, Finnish prefers the partitive. Pouring coffee is seen as dealing with some quantity of a liquid, not a single whole object.

So kahvia is the natural form here; kahvi as a direct object would be very unusual in this context.

Why is it matolle and not matolla or mattoon?

Matolle is the allative case, which basically means “onto the rug” (movement towards a surface).

matto = rug (basic form)
matolla (adessive) = on the rug (location on top, no movement implied)
matolle (allative) = onto the rug (movement onto a surface)
mattoon (illative) = into the rug (movement into an interior)

Here we’re talking about coffee moving onto the surface of the rug, so matolle is the natural choice.

Why is it Älä kaada and not something like sinä älä kaadat?

Älä is the negative imperative for second person singular (“you, one person”). The rules:

Affirmative: Kaada kahvia. = “Pour some coffee.” (2nd person singular imperative)
Negative: Älä kaada kahvia. = “Don’t pour coffee.”

Finnish imperative doesn’t usually mention the subject sinä; it’s understood from the form.
Sinä älä kaada is technically possible but sounds emphatic, like “You don’t pour (it)!” and is less neutral.

Why is the verb form kaada and not kaataa?

The dictionary form is kaataa (to pour / to knock over), but:

In the imperative singular, the form is kaada.
- kaataa → stem kaa
  - da → kaada (imperative stem with -da).

Compare:

antaa (to give) → Anna! (Give!)
tietää (to know) → Tiedä! (Know!)
kaataa (to pour) → Kaada! (Pour!)

So Älä kaada = negative of the imperative “Kaada!”.

What’s the difference between kaataa and kaatua?

They’re related but not the same:

kaataa = a transitive verb: “to pour / to knock something over”
- Kaadan kahvia kuppiin. = I pour coffee into the cup.
- Hän kaatoi tuolin. = He/She knocked the chair over.
kaatua = an intransitive verb: “to fall (over)”
- Tuoli kaatui. = The chair fell over.
- Mies kaatui. = The man fell.

In your sentence, you’re doing something to the coffee, so the verb must be the transitive kaataa → kaada in imperative.

In koska se on vaikea puhdistaa, what does se refer to: the coffee or the rug?

In context, se most naturally refers to matto (“the rug”), even though matto is not repeated in that clause. The idea is:

“Don’t pour coffee on the rug, because it (the rug) is hard to clean.”

Referring to the coffee would be less logical: cleaning the rug is what’s difficult, not the coffee itself.
Finnish often uses se to refer back to a previously mentioned concrete noun that’s obvious from context.

Why is it on vaikea puhdistaa and not something like on vaikea puhdistan or puhdistaa sen?

Puhdistaa here is the basic dictionary form (the 1st infinitive), used in a very common pattern:

on + adjective + verb (1st infinitive)
= “(it) is ADJECTIVE to VERB”

Examples:

On helppo ymmärtää. = It is easy to understand.
On vaikea nähdä. = It is hard to see.
On vaikea puhdistaa. = It is hard to clean.

Because this is an impersonal construction (“it’s hard to clean” in general), the verb puhdistaa does not take a personal ending (puhdistan, puhdistat, etc.).
You could say Sitä on vaikea puhdistaa (“It is hard to clean it”), but then sitä is explicitly the object; in the original, that object is implied from context.

Could this also be koska se on vaikeaa puhdistaa with vaikeaa instead of vaikea?

Yes, vaikea and vaikeaa are both possible here, with a subtle difference:

se on vaikea puhdistaa
- vaikea is in the basic form, acting like a predicative adjective describing se.
- Rough sense: “it is a difficult (thing) to clean.”
se on vaikeaa puhdistaa
- vaikeaa is partitive singular, used more impersonally.
- Rough sense: “cleaning it is difficult” / “it’s difficult to clean (in general).”

In everyday speech, both are acceptable, and many speakers won’t feel a big difference. Your original version with vaikea is very natural.

Why doesn’t Finnish use any words for “the” or “a” here, like “the coffee” or “the rug”?

Finnish has no articles (no direct equivalents of a/an or the). Specificity and definiteness are usually shown by:

Context: You already know which rug is being talked about.
Case forms and word order, and sometimes pronouns.

So:

kahvia can mean “some coffee / any coffee”.
matolle is “onto the rug” (a particular one understood from context, like the rug in the room you’re in).

There is simply no separate word for “the” or “a” to add.

Could you say Älä kaada kahvia matolle, se on vaikea puhdistaa without koska?

Yes, you could say:

Älä kaada kahvia matolle, se on vaikea puhdistaa.

This would sound like two separate statements next to each other:

“Don’t pour coffee on the rug,”
“it’s hard to clean.”

The cause–effect relationship is still clear from context, but koska makes the reason explicit:

… koska se on vaikea puhdistaa. = “… because it’s hard to clean.”

Both are correct; using koska just emphasizes that the second part is specifically the reason for the command.