Tvättkorgen står i badrummet, och den är redan full.

Because tvättkorgen is the definite form: the laundry basket.

In Swedish, the definite article is usually added as an ending to the noun:

• en tvättkorg = a laundry basket
• tvättkorgen = the laundry basket

So instead of using a separate word like English the, Swedish often adds -en or -et to the noun.

For the same reason: badrummet is the definite form, meaning the bathroom.

• ett badrum = a bathroom
• badrummet = the bathroom

So i badrummet means in the bathroom.

Notice that badrum is an ett-word, so its definite ending is -et:

• ett badrum
• badrummet

Den means it for a singular en-word noun.

The noun tvättkorg is an en-word:

• en tvättkorg
• so later you refer to it as den

If the noun were an ett-word, you would normally use det instead.

So in this sentence:

• Tvättkorgen står i badrummet, och den är redan full.

den refers back to tvättkorgen, not badrummet.

That is also clear from gender:

• tvättkorg = en-word → den
• badrum = ett-word → det

Because of grammatical gender.

• tvättkorg is an en-word
• badrum is an ett-word

The sentence uses den, which matches an en-word, so it must refer to tvättkorgen.

If it referred to badrummet, Swedish would normally use det instead:

• Badrummet ... och det är redan fullt.

Because the adjective agrees with the noun it describes.

Here, full describes tvättkorgen, which is an en-word in singular form. For singular en-words, the basic adjective form is used:

• en korg är full = a basket is full

Compare:

• ett rum är fullt = a room is full

So:

• den är redan full = correct, because den refers to tvättkorgen
• det är redan fullt would fit an ett-word noun instead

Swedish often uses a position verb where English would simply use is.

stå literally means stand, but it is also commonly used for objects that are upright or placed somewhere in a standing position.

So:

• Tvättkorgen står i badrummet
  literally: The laundry basket stands in the bathroom
• natural English meaning: The laundry basket is in the bathroom

This is very common in Swedish:

• Boken ligger på bordet = The book is on the table
  (literally lies)
• Tavlan hänger på väggen = The picture is on the wall
  (literally hangs)

So står sounds natural because a laundry basket is imagined as standing upright somewhere.

Redan means already.

In a normal Swedish main clause, sentence adverbs like redan usually come after the finite verb:

• den är redan full

That is the standard word order:

• subject: den
• finite verb: är
• adverb: redan
• adjective/complement: full

This is similar to other Swedish sentence adverbs:

• Han är aldrig sen = He is never late
• Vi har redan ätit = We have already eaten

You may also hear full redan in some contexts, but är redan full is the neutral, standard placement.

Because Swedish forms compounds very freely, much more often than English does.

• tvätt = washing / laundry
• korg = basket
• tvättkorg = laundry basket

And:

• bad = bath
• rum = room
• badrum = bathroom

In Swedish, compound nouns are normally written as one word, not two.

Also, the last part of the compound determines the grammatical gender:

• korg is an en-word → en tvättkorg
• rum is an ett-word → ett badrum

Then the definite ending is added to the whole compound:

• tvättkorgen
• badrummet

Because Swedish usually expresses definiteness with an ending on the noun instead of a separate article.

So:

• the laundry basket = tvättkorgen
• the bathroom = badrummet

A separate definite word like English the is usually not needed.

Very roughly:

• English: the basket
• Swedish: basket-the

There are cases in Swedish where a separate definite word is used too, especially with adjectives:

• den fulla tvättkorgen = the full laundry basket

But in your sentence, with just the noun by itself, the ending is enough.

Not always.

Swedish often uses fewer commas than English, and many writers would simply write:

• Tvättkorgen står i badrummet och den är redan full.

The comma is not wrong here, but it is often optional before och when linking two main clauses.

So both versions may be seen, depending on style.

Yes, but that would change what it refers to.

In your sentence:

• den är redan full = it is already full
• den refers to tvättkorgen

If you wanted to talk about the bathroom instead, you would use det, because badrum is an ett-word:

• Badrummet är redan fullt = The bathroom is already full

So den/full and det/fullt are not interchangeable here; they depend on the gender of the noun being described.