Ich muss den Kissenbezug noch waschen, bevor ich schlafen gehe.

Because Kissenbezug is the direct object of waschen.

• The basic dictionary form is der Kissenbezug.
• Der is nominative masculine.
• After a verb like waschen, the thing being washed is in the accusative.
• Masculine nouns change from der to den in the accusative.

So:

• der Kissenbezug = the pillowcase, as the subject
• den Kissenbezug = the pillowcase, as the object

German very often combines nouns into a single compound noun.

Here:

• Kissen = pillow
• Bezug = cover / case

So Kissenbezug literally means something like pillow cover, i.e. pillowcase.

A useful rule: in German compounds, the last part determines the grammatical gender. Since Bezug is masculine (der Bezug), the whole word is also masculine: der Kissenbezug.

In this sentence, noch means something like still, yet, or still have to.

It gives the idea that this is one more thing that remains to be done before sleeping.

So Ich muss den Kissenbezug noch waschen suggests:

• I still need to wash the pillowcase.
• I haven’t done it yet, but I need to.

This is a very common use of noch with tasks:

• Ich muss noch einkaufen.
• Wir müssen noch aufräumen.

Because müssen is a modal verb.

With modal verbs in German:

• the modal verb is conjugated
• the main verb stays in the infinitive
• that infinitive usually goes to the end of the clause

So:

• Ich muss ... waschen.

That is the normal pattern:

• Ich muss arbeiten.
• Sie kann kommen.
• Wir wollen essen.

In your sentence, muss is in second position, and waschen goes to the end of the main clause.

Because bevor introduces a subordinate clause.

In German, subordinate clauses are normally separated by a comma from the main clause.

So the structure is:

• Ich muss den Kissenbezug noch waschen
• bevor ich schlafen gehe

This comma is not optional in standard German.

Because bevor is a subordinating conjunction, and in German subordinate clauses the conjugated verb goes to the end.

So:

• main clause: Ich gehe schlafen.
• subordinate clause: ..., bevor ich schlafen gehe.

This happens with many subordinating conjunctions, such as:

• weil
• dass
• wenn
• obwohl
• bevor

Example:

• Ich bleibe zu Hause, weil ich krank bin.
• Sie kommt, wenn sie Zeit hat.

Schlafen gehen is a very common German verb combination meaning to go to sleep or to go to bed to sleep.

Literally it looks like to go sleep, but idiomatically it is completely normal German.

Compare:

• Ich gehe schlafen. = I’m going to sleep / going to bed
• Ich schlafe. = I am sleeping

So bevor ich schlafen gehe means before I go to sleep, not before I sleep in the general sense.

That difference matters:

• bevor ich schlafe sounds more like before I am sleeping / before I sleep
• bevor ich schlafen gehe means before I go off to sleep

There are two reasons here.

1. After a modal verb, you do not use zu
With müssen, können, wollen, sollen, dürfen, and mögen, the other verb appears as a bare infinitive.

So:

• Ich muss waschen.
• not Ich muss zu waschen.

2. In the expression schlafen gehen, gehen combines with another infinitive without zu
This is similar to expressions like:

• einkaufen gehen
• spazieren gehen
• tanzen gehen

So:

• Ich gehe schlafen.
• not Ich gehe zu schlafen.

Yes. German word order is somewhat flexible, especially with adverbs like noch and with objects. The original sentence is very natural, but other versions are possible.

For example:

• Ich muss den Kissenbezug noch waschen.
  Neutral, very natural.

• Ich muss noch den Kissenbezug waschen.
  Also natural; noch is placed a bit earlier.

• Den Kissenbezug muss ich noch waschen.
  This puts extra emphasis on the pillowcase.

The exact placement can slightly change the focus, but the core meaning stays the same.

They are close, but not exactly the same.

• bevor ich schlafen gehe = before I go to sleep
• bevor ich ins Bett gehe = before I go to bed

Going to bed does not always mean going straight to sleep. You might go to bed and then read, use your phone, or talk to someone first.

So schlafen gehen focuses more directly on sleeping, while ins Bett gehen focuses on going to bed.

In many everyday situations, though, the two are very similar.