Nach dem Waschen lege ich den Kissenbezug auf den Wäscheständer, wobei das Waschmittel noch neben der Maschine steht.

Because nach takes the dative case in this meaning of after.

• das Waschen = the washing / washing
• dative of das Waschen = dem Waschen

So:

• nach dem Waschen = after washing

This is a very common pattern in German:

• nach dem Essen = after eating / after the meal
• nach der Arbeit = after work
• nach dem Unterricht = after class

Here, waschen is being used as a noun, not as a normal verb. In German, nouns are capitalized.

• verb: waschen = to wash
• noun: das Waschen = washing

So nach dem Waschen literally uses the verbal idea washing as a noun phrase.

Because Nach dem Waschen is placed first, and German main clauses follow the verb-second rule.

The first position is occupied by:

• Nach dem Waschen

So the finite verb must come next:

• lege

Then comes the subject:

• ich

That gives:

• Nach dem Waschen lege ich ...

If you started with the subject instead, you could also say:

• Ich lege nach dem Waschen den Kissenbezug auf den Wäscheständer ...

Both are possible, but the original version emphasizes after washing.

Because den Kissenbezug is the direct object of lege.

The noun is:

• der Kissenbezug = the pillowcase

In the accusative case, masculine der becomes den:

• nominative: der Kissenbezug
• accusative: den Kissenbezug

Since you are placing the pillowcase, it is the thing directly affected by the action.

Kissenbezug means pillowcase.

It is a compound noun:

• Kissen = pillow / cushion
• Bezug = cover

So literally it is a pillow cover.

In everyday use, Kissenbezug is natural German for pillowcase.

Because auf is a two-way preposition, and here it shows movement toward a destination, so German uses the accusative.

• auf den Wäscheständer legen = to place it onto the drying rack
• auf dem Wäscheständer liegen = to be lying on the drying rack

So the contrast is:

• Ich lege den Kissenbezug auf den Wäscheständer.
  I put the pillowcase onto the drying rack.
• Der Kissenbezug liegt auf dem Wäscheständer.
  The pillowcase is lying on the drying rack.

Movement/change of location -> accusative
Location/state -> dative

Der Wäscheständer is a clothes drying rack or airer.

It is made from:

• Wäsche = laundry
• Ständer = stand / rack

So it is the rack where clothes are hung or laid to dry.

Depending on region and context, English translations might be:

• drying rack
• clothes rack
• laundry rack
• clothes airer

Here wobei introduces additional background information happening at the same time. In this sentence, it is roughly like:

• while
• with
• at the same time that
• and meanwhile

So:

• ..., wobei das Waschmittel noch neben der Maschine steht.

means something like:

• ..., while the detergent is still standing next to the machine.

It does not mean the detergent is necessary for putting the pillowcase on the rack. It simply adds a side detail to the scene.

Because wobei introduces a subordinate clause, and in German subordinate clauses the finite verb usually goes to the end.

Main clause:

• Nach dem Waschen lege ich den Kissenbezug auf den Wäscheständer

Subordinate clause:

• wobei das Waschmittel noch neben der Maschine steht

That is why steht comes at the end of the wobei clause.

German often uses position verbs more literally than English.

• stehen = stand
• liegen = lie
• sitzen = sit

A bottle, box, or container of detergent is often thought of as standing upright, so German says:

• Das Waschmittel steht neben der Maschine.

English often just says:

• The detergent is next to the machine.

If it were lying on its side, German might use liegt instead.

Here noch means still.

• das Waschmittel noch neben der Maschine steht
• the detergent is still next to the machine

It suggests that the detergent remains there and has not been put away yet.

Common meanings of noch include:

• still
• yet
• another / one more
• sometimes even

In this sentence, still is the right idea.

Because the context already makes it clear that Maschine means the washing machine.

The fuller word would be:

• die Waschmaschine = washing machine

But native speakers often shorten it to:

• die Maschine

when the context is obvious, especially in household situations.

So both are possible:

• neben der Maschine
• neben der Waschmaschine

The longer version is clearer if there could be any ambiguity.

Because in German, subordinate clauses are separated by a comma.

Since wobei introduces a subordinate clause, a comma is required:

• ..., wobei das Waschmittel noch neben der Maschine steht.

This is standard German punctuation.

Usually not in natural everyday English. Although wobei can sometimes correspond to whereby in very formal or logical contexts, here whereby would sound unnatural.

Better translations in this sentence are:

• while
• with
• as
• and meanwhile

So for this sentence, while is probably the most natural English choice.

Yes, lege is a good choice because legen means to lay / put something down. It is often used when placing something soft, flat, or horizontal.

• legen = lay, put down
• stellen = stand something upright
• setzen = set something into a sitting position, or seat someone/something

A pillowcase is something you would naturally legen, not stellen.

So:

• Ich lege den Kissenbezug auf den Wäscheständer.

sounds natural.

Yes, it is grammatical and understandable. However, the wobei part adds a somewhat descriptive side detail, so it sounds a bit more like written or carefully described language than a very simple everyday statement.

A simpler version might be:

• Nach dem Waschen lege ich den Kissenbezug auf den Wäscheständer. Das Waschmittel steht noch neben der Maschine.

But the original sentence is perfectly fine if the goal is to connect the two ideas smoothly.

There are several useful case examples here:

• nach dem Waschen -> dative after nach
• den Kissenbezug -> accusative as the direct object
• auf den Wäscheständer -> accusative because auf shows movement toward a destination
• neben der Maschine -> dative because neben here describes location, not movement

So the sentence is a good example of how German cases depend on both the verb and the prepositions used.