Eine Wäscheklammer fehlt noch, deshalb bleibt das Handtuch erst einmal über der Tür hängen.

Wäscheklammer means clothespin or clothes peg.

It is a compound noun:

• Wäsche = laundry / washing
• Klammer = clip, clamp, fastener

So the literal idea is a laundry clip.

Like most German compounds, the last part determines the grammatical gender:

• die Klammer
• therefore die Wäscheklammer

That is why the sentence begins with Eine Wäscheklammer.

Here, fehlen means to be missing or to be lacking.

This verb often works differently from English to miss, which can confuse learners.

In German:

• Eine Wäscheklammer fehlt. = A clothespin is missing.
• Mir fehlt eine Wäscheklammer. = I’m missing a clothespin / I don’t have a clothespin.

So the thing that is absent is often the subject in German.

In this sentence:

• Eine Wäscheklammer fehlt noch = A clothespin is still missing

Noch here means still or yet.

So:

• Eine Wäscheklammer fehlt. = A clothespin is missing.
• Eine Wäscheklammer fehlt noch. = A clothespin is still missing / One clothespin is yet to be found.

It adds the idea that the situation has not been resolved yet.

Because German main clauses follow the verb-second rule.

That means the finite verb must come in the second position of the clause.

In the second clause:

• deshalb is in first position
• so the verb bleibt must come next

That gives:

• Deshalb bleibt das Handtuch ...

Not:

• Deshalb das Handtuch bleibt ...

A helpful pattern is:

• Subject first: Das Handtuch bleibt deshalb ...
• Adverb first: Deshalb bleibt das Handtuch ...

Both are possible, but the verb stays in second position.

Erst einmal here means for now, for the time being, or until further notice.

It suggests a temporary solution:

• there is no clothespin yet,
• so the towel will stay over the door for now.

It does not literally mean first once in normal English. It is an idiomatic expression.

Similar translations:

• erst einmal = for the moment
• erst mal = shorter, more conversational form

Because über is a two-way preposition:

• it takes the dative for location
• it takes the accusative for movement toward a destination

Here the towel is already in a position over the door, so this is location:

• über der Tür = over the door

Compare:

• Das Handtuch hängt über der Tür. = The towel is hanging over the door.
• Ich hänge das Handtuch über die Tür. = I hang the towel over the door.

So:

• where? → dative
• to where? → accusative

This is a common German verb combination meaning to remain hanging or to stay hanging.

The idea is:

• hängen = to hang
• bleiben = to remain, stay

Together:

• hängen bleiben = to stay hanging / remain hanging

In a main clause, the finite verb goes near the front, and the other verb goes to the end:

• Das Handtuch bleibt ... hängen.

So the structure is very natural in German.

In this sentence, it means that the towel will continue to hang there because there is no clothespin yet.

Here it is intransitive, meaning there is no direct object of hängen.

German actually has two closely related uses:

• etwas hängen = to hang something up
• hängen = to be hanging

Compare:

• Ich hänge das Handtuch über die Tür. = I hang the towel over the door.
• Das Handtuch hängt über der Tür. = The towel is hanging over the door.

In your sentence, the towel is not being actively hung up at that moment. It is simply remaining in that hanging position:

• bleibt ... hängen

Because das Handtuch is the subject of the second clause.

The second clause is:

• deshalb bleibt das Handtuch erst einmal über der Tür hängen

Ask what remains hanging?

• das Handtuch

That makes it the subject, so it is in the nominative.

Yes. Common alternatives are:

• darum
• daher
• deswegen

For example:

• Eine Wäscheklammer fehlt noch, deshalb bleibt das Handtuch ...
• Eine Wäscheklammer fehlt noch, deswegen bleibt das Handtuch ...

They all mean something like therefore / that’s why / for that reason.

There can be small style differences, but in a sentence like this they are very close in meaning.

Yes. Erst mal is a very common shorter form of erst einmal.

So these are both natural:

• ... bleibt das Handtuch erst einmal über der Tür hängen.
• ... bleibt das Handtuch erst mal über der Tür hängen.

The full form erst einmal may sound a little more neutral or careful.
The shorter erst mal is very common in everyday spoken German.