Der Kissenbezug ist noch nicht trocken, aber die Bettwäsche kann schon in den Schrank.

Kissenbezug is a masculine noun, so it takes der in the nominative singular: der Kissenbezug.

The word is a compound:

• das Kissen = pillow
• der Bezug = cover

In German compounds, the last part usually determines the gender, so because Bezug is masculine, Kissenbezug is also masculine.

Bettwäsche is a singular noun in German, even though in English you might translate it as bedding or sometimes bed linens/sheets.

It is a collective noun:

• die Bettwäsche = bed linen / bedding
• grammatically singular

That is why the sentence says:

• die Bettwäsche kann ... not
• die Bettwäsche können ...

So the grammar follows the singular noun, even if the idea includes several items.

Because trocken is being used predicatively, after the verb sein.

In German:

• ein trockener Kissenbezug = a dry pillowcase
  → adjective before a noun, so it gets an ending
• Der Kissenbezug ist trocken = The pillowcase is dry
  → adjective after sein, so it does not get an ending

That is a very common pattern:

• Das Wasser ist kalt
• Die Tür ist offen
• Der Boden ist sauber

Nicht trocken just means not dry.

Noch nicht trocken means not dry yet.

So noch adds the idea that the situation is expected to change later.

Compare:

• Der Kissenbezug ist nicht trocken. = The pillowcase is not dry.
• Der Kissenbezug ist noch nicht trocken. = The pillowcase is not dry yet.

In this sentence, noch nicht works well because the pillowcase is expected to become dry later.

German main clauses usually put the finite verb in second position. Here, the finite verb is ist.

So the structure is:

• Der Kissenbezug = subject
• ist = finite verb
• noch nicht trocken = rest of the predicate

The sentence follows normal main-clause word order:

• Der Kissenbezug ist noch nicht trocken

Also, nicht comes before the adjective trocken, because it is negating the idea of being dry.

This is a very natural kind of ellipsis in German. A fuller version would be something like:

• ... aber die Bettwäsche kann schon in den Schrank gelegt werden.
• ... aber die Bettwäsche kann schon in den Schrank.

The shorter version leaves out something that is easy to understand from context, such as:

• gelegt werden = be put
• geräumt werden = be put away
• verstaut werden = be stored

English does this too sometimes:

• The laundry can already go in the closet.

So although kann is normally used with another verb, everyday German often omits that verb when the meaning is obvious.

Because this sentence expresses movement/direction into the cupboard or closet, not location inside it.

With in, German uses:

• accusative for direction/movement
• dative for location

So:

• in den Schrank = into the cupboard/closet
• im Schrank (= in dem Schrank) = in the cupboard/closet

Here the bedding is being put away, so it is moving into the closet:

• die Bettwäsche kann schon in den Schrank

Here schon means already.

It creates a contrast with noch nicht in the first clause:

• noch nicht trocken = not dry yet
• schon in den Schrank = can already go into the closet

So the sentence contrasts two different states:

• the pillowcase is not ready yet
• the bedding is ready already

That contrast is one of the main reasons schon is used here.

Because aber is a coordinating conjunction. Coordinating conjunctions do not send the verb to the end.

So you get:

• Der Kissenbezug ist noch nicht trocken, aber die Bettwäsche kann schon in den Schrank.

Compare that with a subordinating conjunction like weil:

• ..., weil der Kissenbezug noch nicht trocken ist.

So after aber, German keeps normal main-clause word order:

• subject + finite verb + rest

Yes. A more explicit version would be:

• Der Kissenbezug ist noch nicht trocken, aber die Bettwäsche kann schon in den Schrank gelegt werden.

That sounds more complete and slightly more formal.

The original version is shorter and very natural in everyday German:

• ... kann schon in den Schrank.

Native speakers often prefer this shorter version when the missing verb is obvious from context.

Schrank is a general word for a large storage unit such as a:

• cupboard
• cabinet
• closet
• wardrobe

The exact English translation depends on context.

In this sentence, in den Schrank probably means something like:

• into the closet
• into the cupboard
• into the wardrobe

If it is about storing clean bedding, closet or cupboard is often the best natural translation.