Der Stock liegt neben der Tür.

Because the two der forms are doing different jobs.

• der Stock is nominative singular masculine because it is the subject of the sentence.
• der Tür is dative singular feminine because it comes after neben in a location phrase.

The noun Tür is normally:

• die Tür = nominative/accusative singular
• der Tür = dative/genitive singular

So in this sentence:

• Der Stock = the stick/cane
• neben der Tür = next to the door

This is a very common thing in German: the article changes based on case, not just gender.

Neben is a two-way preposition (Wechselpräposition). That means it can take either:

• dative for location: where something is
• accusative for direction/movement toward something

In Der Stock liegt neben der Tür, the stick is already there, so this is location, and German uses the dative:

• neben der Tür = next to the door

Compare:

• Der Stock liegt neben der Tür. = The stick is lying next to the door.
• Ich lege den Stock neben die Tür. = I put the stick next to the door.

In the second sentence, there is movement toward that position, so neben takes the accusative: die Tür.

German often prefers a more specific verb for position.

Here, liegt comes from liegen, which means to lie or to be lying. It is used for things that are resting in a horizontal position, or more generally for things that are located somewhere in a way German conceptualizes as lying.

So:

• Der Stock liegt neben der Tür.

sounds natural because a stick is typically imagined as lying on the ground or floor.

German often distinguishes between:

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

English often just says is in these cases, but German usually uses one of these position verbs.

A useful beginner rule is:

• use liegen for things that are lying flat, spread out, or not upright
• use stehen for things that are upright or standing on a base

So:

• Das Buch liegt auf dem Tisch. = The book is lying on the table.
• Die Flasche steht auf dem Tisch. = The bottle is standing on the table.

For Stock, German normally imagines it as lying somewhere unless the context clearly suggests it is upright.

So Der Stock liegt neben der Tür is the natural choice.

Der Stock is nominative singular.

Why?

Because it is the subject of the sentence — the thing doing the verb liegt.

You can identify the basic structure like this:

• Der Stock = subject
• liegt = verb
• neben der Tür = prepositional phrase showing location

So the sentence follows the standard pattern:

• Subject + verb + location

Yes. German word order is flexible, but the finite verb usually stays in the second position in a main clause.

So these are both correct:

• Der Stock liegt neben der Tür.
• Neben der Tür liegt der Stock.

The second version emphasizes the location a bit more.

What does not change is that liegt stays in second position in a normal main clause.

Here, Stock means something like:

• stick
• cane
• staff

German words often have more than one meaning depending on context. Stock can also mean floor/storey in some contexts, but not here.

In Der Stock liegt neben der Tür, the context clearly points to the physical object meaning: a stick/cane.

Because in German, all nouns are capitalized.

So:

• der Stock
• die Tür

are both written with capital letters because Stock and Tür are nouns.

This is one of the most noticeable spelling rules in German, and it applies all the time, not just at the beginning of a sentence.

liegt is pronounced roughly like leegt in English approximation.

A few details:

• ie in German usually sounds like a long ee
• so liegt sounds approximately leekt, but with a German g/t ending
• the g in this word is not pronounced like the English g in go

A rough learner-friendly pronunciation would be:

• leekt

But if you want to be more accurate, it is closer to leehkt with a clear final t sound.

People would probably understand you, but it is less idiomatic than liegt.

German often prefers the more precise location verb:

• liegt for lying
• steht for standing
• sitzt for sitting

So Der Stock liegt neben der Tür sounds more natural than Der Stock ist neben der Tür.

Using ist is not always strictly wrong, but it often sounds less native-like when German expects a position verb.