Learn Hardcore German: Das Möbelstück passt nicht durch die Tür, deshalb müssen wir es durch das Fenster tragen. - The piece of furniture doesn’t fit through the door, so we have to carry it through the window.

Möbelstück is a neuter noun in German, so its nominative singular article is das: das Möbelstück.
Also, Möbel by itself is usually used as a plural-only noun (die Möbel = furniture), but Möbelstück is a countable singular item (a piece of furniture).

What does passen mean here, and how is it different from passen zu?

Here passen means to fit (physically), often used with places/openings: durch die Tür passen = to fit through the door.
passen zu means to match / go with in a stylistic or suitability sense: Das passt zu dir = That suits you / That matches you.

Why is it passt and not passen?

passt is the 3rd person singular present tense form of passen, agreeing with the subject das Möbelstück (it):

ich passe
du passt
er/sie/es passt
wir passen
ihr passt
sie/Sie passen

Why is nicht placed before durch die Tür?

nicht usually negates what comes after it (or the whole statement depending on position).
In Das Möbelstück passt nicht durch die Tür, the negation targets the idea of fitting through the door: it does not fit through that opening. Placing nicht right before the prepositional phrase is a very natural choice here.

Why does durch take die Tür / das Fenster (accusative), and how can I tell?

durch is a preposition that always takes the accusative case. So you get:

durch die Tür (Tür is feminine; accusative looks the same as nominative here)
durch das Fenster (Fenster is neuter; accusative also looks the same as nominative here)
You can tell mainly by memorizing that durch + Akkusativ is fixed.

Why is there a comma before deshalb?

Because the sentence contains two main clauses: 1) Das Möbelstück passt nicht durch die Tür
2) deshalb müssen wir es durch das Fenster tragen
German commonly separates two independent main clauses with a comma, especially when the second one begins with a connector like deshalb.

What exactly is deshalb, and how does it affect word order?

deshalb means therefore / that’s why and it functions as an adverb connecting the logic between clauses.
When deshalb comes first in its clause, it takes the first position, and the conjugated verb still has to be in second position (V2 rule). That’s why you get:
deshalb müssen wir ...
(not deshalb wir müssen ...)

Could I replace deshalb with weil, and what would change?

Yes, but the structure changes because weil introduces a subordinate clause with the verb at the end. For example:
Wir müssen es durch das Fenster tragen, weil das Möbelstück nicht durch die Tür passt.
With deshalb you have two main clauses (V2 word order). With weil the reason clause is subordinate (verb-final).

Why is müssen in second position, and why is tragen at the end?

müssen is the conjugated (finite) verb in the clause, so it goes in second position in a main clause: deshalb müssen wir ...
tragen is an infinitive dependent on the modal verb müssen, and German puts infinitives at the end of the clause: ... tragen.

What does es refer to, and why is it es?

es refers back to das Möbelstück. Since Möbelstück is neuter, the matching pronoun is es (it).
It’s the direct object of tragen (we have to carry it), so es is accusative here—conveniently, es looks the same in nominative and accusative.

Why do we say durch das Fenster tragen (through the window) instead of something like zum Fenster?

durch focuses on the path through an opening: you’re moving the object from outside to inside (or vice versa) via the window opening.
If you said zum Fenster tragen, that would mean carry it to the window (ending at the window, not necessarily passing through it).

Are there more natural alternatives to Möbelstück in everyday German?

Yes. Depending on context, people often use more specific nouns: der Schrank (cabinet), das Sofa, der Tisch, etc.
Or they may say something like Das Teil passt nicht durch die Tür (This thing/part doesn’t fit through the door) when the exact item is obvious.