Hvis kurven er fuld, tager jeg vasketøjet med ned i vaskeriet.

Hvis means if and introduces a condition:

• Hvis kurven er fuld = If the basket is full

It is the normal word to use for real conditions in Danish, just like if in English.

This is a very common Danish word order rule.

In a main clause, the finite verb usually comes in second position. When the sentence starts with something other than the subject—here, the whole if-clause—the verb comes before the subject in the main clause.

So:

• Hvis kurven er fuld, tager jeg vasketøjet med ned i vaskeriet.

Structure:

• Hvis kurven er fuld = introductory clause
• tager = verb in second position
• jeg = subject

If you started with the subject instead, you would say:

• Jeg tager vasketøjet med ned i vaskeriet, hvis kurven er fuld.

Both are correct.

Danish often uses the present tense to talk about the future, especially when the meaning is clear from context.

So:

• tager jeg vasketøjet med ned i vaskeriet
  literally looks like I take the laundry down to the laundry room
• but in context it means I’ll take the laundry down to the laundry room

English also does something similar sometimes:

• If it rains, I stay home.
• Tomorrow I leave early.

So this is very natural Danish.

Kurv means basket.
Kurven means the basket.

Danish often makes nouns definite by adding the article to the end of the noun:

• en kurv = a basket
• kurven = the basket

So Hvis kurven er fuld literally means If the basket is full.

Fuld is the adjective full.

Here it comes after er in kurven er fuld, so it is a predicative adjective. In that position, adjectives often stay in their basic/common-gender singular form when the subject is a common-gender noun like kurven.

Compare:

• Kurven er fuld = The basket is full
• Huset er fuldt = The house is full
  because hus is a neuter noun
• Kurvene er fulde = The baskets are full

So the form depends on the noun it describes.

Vasketøj means laundry / washing.
Vasketøjet means the laundry.

Again, Danish adds the definite article to the end:

• vasketøj = laundry
• vasketøjet = the laundry

This word is often used like an uncountable noun in English, similar to laundry rather than clothes in this context.

In tager vasketøjet med, the med is part of the expression tage noget med, which means:

• to take something along
• to bring something with you

So:

• Jeg tager vasketøjet med = I take/bring the laundry with me

Without med, tage just means take in a more general sense. With med, it clearly suggests bringing something along somewhere.

They do different jobs:

• med = along / with
• ned = down

So:

• tager jeg vasketøjet med = I take the laundry with me
• ned i vaskeriet = down into the laundry room

Together, med ned gives the idea take it down with me.

This kind of combination is very common in Danish.

Good question. The difference is roughly:

• i = in / into
• til = to / toward

Vaskeriet here usually means the laundry room or laundromat as a place you go into, so ned i vaskeriet suggests movement into that space:

• down into the laundry room

If you said til vaskeriet, it would focus more on going to the place rather than into it. In this sentence, i sounds natural because the speaker is taking the laundry down to the room itself.

Vaskeri means laundry room, laundromat, or washing facility, depending on context.
Vaskeriet means the laundry room / the laundromat.

So:

• et vaskeri = a laundry room / a laundromat
• vaskeriet = the laundry room / the laundromat

In many everyday contexts, especially in apartment buildings, vaskeriet means the shared laundry room in the building.

Because the sentence begins with a subordinate clause:

• Hvis kurven er fuld

and then moves to the main clause:

• tager jeg vasketøjet med ned i vaskeriet

In standard Danish, a comma is often used to separate these parts. So the comma helps mark the boundary between the if-clause and the main statement.

Yes, but the meaning changes slightly.

• Hvis kurven er fuld = If the basket is full
  This is a condition. Maybe it is full, maybe not.

• Når kurven er fuld = When the basket is full
  This sounds more like you expect it to happen, or you are talking about a regular situation.

So if the sentence is about a possible condition, hvis is better. If it is about a repeated or expected event, når may fit better.