Learn Hardcore German: Nach dem Umzug fehlt ein Möbelstück, und ich finde es erst im letzten Karton. - After the move, a piece of furniture is missing, and I only find it in the last box.

In this sentence, der Umzug means the move / moving house (the event/process).
It can also be used in related senses depending on context (e.g., the whole moving operation), but here it’s clearly after moving.

Why does German say fehlt ein Möbelstück instead of something like ist weg?

fehlen is a very common way to express something is missing. It’s slightly more “neutral/factual” than weg sein (to be gone).
Structure-wise, German often uses:

X fehlt. = X is missing.
Here: ein Möbelstück fehlt = a piece of furniture is missing.

How does fehlen work grammatically? Who/what is the subject?

The thing that is missing is the grammatical subject.

ein Möbelstück = subject (nominative)
There’s no direct object here. If you add from whom, you can use a dative:
Mir fehlt ein Möbelstück. = I’m missing a piece of furniture / A piece of furniture is missing (for me).
In your sentence, the speaker is implied by context, so mir is omitted.

Why is it ein Möbelstück and not ein Möbel?

Möbel is usually a collective/plural-ish word in German (often used like furniture or pieces of furniture).
To talk about one item, German commonly uses:

ein Möbelstück = a piece/item of furniture
You can say ein Möbel in some contexts, but it’s less standard/less common than ein Möbelstück for “one item.”

What does Möbelstück literally mean, and is it always one “piece”?

Literally Möbel (furniture) + Stück (piece) = piece of furniture.
It generally refers to a single item like a chair, table, shelf, dresser, etc.

What’s the role of und here—could this be two independent sentences?

Yes, it’s essentially two main clauses joined with und: 1) Nach dem Umzug fehlt ein Möbelstück,
2) und ich finde es erst im letzten Karton.
German often connects main clauses with a comma + und, especially when the clauses are longer.

Why is the verb in the second part finde and not at the end?

Because the second part is a main clause (ich finde ...). In German main clauses, the finite verb is in position 2 (V2 rule):

ich (position 1) + finde (position 2) + rest

If it were a subordinate clause (e.g., with weil), the verb would go to the end:

..., weil ich es erst im letzten Karton finde.

What does erst mean here? Is it “first” or “only”?

Here erst means only / not until (then)—it expresses that something happens later than expected:

ich finde es erst im letzten Karton = I only find it in the last box / I don’t find it until the last box.

It can also mean first in other contexts (e.g., erstens, erst mal), but here it’s the “not until” meaning.

Why is it im letzten Karton and not in dem letzten Karton?

im is just a contraction of in dem.

in dem letzten Karton = im letzten Karton
Both are correct; im is more natural in everyday German.

Also, in with a location (not motion) takes dative, hence dem:

der Karton → dem Karton (dative)

Why is it letzten (with -en) and not letzte?

Because letzten is an adjective ending used with a masculine dative noun after dem:

in dem/im
- Karton (masculine) → dative: dem Karton
  Adjective ending in this pattern: dem letzten Karton.

What does es refer to? Is it the missing item?

Yes. es refers back to ein Möbelstück.
German often uses es for a previously mentioned noun, even if in English you might prefer it (which matches perfectly here anyway).

Could you say Nach dem Umzug fehlt mir ein Möbelstück...? Would that sound better?

Yes, that’s very natural and often what native speakers would say, because it explicitly marks the “person affected”:

Nach dem Umzug fehlt mir ein Möbelstück, und ich finde es erst im letzten Karton.
Your original version is still correct; it just keeps the sentence a bit more “impersonal.”

Does Karton mean “cardboard” or “box”?

der Karton can mean a cardboard box/carton. In moving contexts, (Umzugs-)Karton is the standard word for a moving box.
So im letzten Karton = in the last box (i.e., the last moving box you open).

Is there a difference between Nach dem Umzug and Nach dem Umziehen?

Yes:

Nach dem Umzug = after the move (noun: the moving event)
Nach dem Umziehen = after moving (verbal noun/gerund-like)
Both can work, but Nach dem Umzug is very idiomatic and common.