Der Nachziehstapel liegt neben der Kartenauslage.

They are both forms of the definite article the, but in different grammatical roles.

• Der Nachziehstapel is the subject of the sentence, so it is in the nominative case.
• neben der Kartenauslage uses der because neben here takes the dative case when it describes a location.

So:

• der Nachziehstapel = the draw pile (nominative, masculine)
• der Kartenauslage = next to the card display/layout (dative, feminine)

Even though the article looks the same in English, German changes it depending on gender + case.

Because this sentence describes where something is, not where something is going.

German prepositions like neben can take either:

• dative for location: next to
• accusative for movement toward: to next to / beside

Here, liegt means is lying / is located, so this is a static location. That is why German uses the dative:

• neben der Kartenauslage = next to the card display

Compare:

• Der Nachziehstapel liegt neben der Kartenauslage.
  The draw pile is next to the card display.
• Ich lege den Nachziehstapel neben die Kartenauslage.
  I put the draw pile next to the card display.

German often prefers specific position verbs where English might simply use is.

liegen means:

• to lie
• to be lying
• by extension, to be situated / to be located for something that is flat or resting horizontally

A pile of cards is understood as lying flat, so liegt sounds natural.

Compare some common German position verbs:

• liegen = to lie / be lying
• stehen = to stand / be standing
• sitzen = to sit / be seated
• hängen = to hang

So Der Nachziehstapel liegt neben der Kartenauslage is more idiomatic than ist here.

Yes. Nachziehstapel is a compound noun, which is extremely common in German.

It is built from:

• nachziehen = to draw more / draw after / replenish by drawing, depending on context
• Stapel = stack / pile

In a card-game context, Nachziehstapel means the draw pile: the pile you draw cards from.

German often combines several ideas into one noun instead of using separate words like English does.

It is also a compound noun:

• Karten = cards
• Auslage = display, layout, spread, items laid out

In a card-game context, Kartenauslage usually means something like:

• the card display
• the layout of cards
• the row/spread of available cards

German writes these compounds as one word, which is normal:

• Karten + Auslage = Kartenauslage

You usually have to learn the gender together with the noun.

In this sentence:

• der Nachziehstapel → masculine
• die Kartenauslage → feminine

That is why you get:

• Der Nachziehstapel in the nominative
• der Kartenauslage in the dative, because feminine die changes to der

A useful habit is to memorize nouns with their article:

• der Nachziehstapel
• die Kartenauslage

not just the bare noun by itself.

This is the normal German main-clause word order:

1. subject: Der Nachziehstapel
2. verb in second position: liegt
3. the rest of the sentence: neben der Kartenauslage

German main clauses usually put the finite verb in position 2.

So this sentence follows a very standard pattern:

• Subject + verb + location

You could also change the emphasis:

• Neben der Kartenauslage liegt der Nachziehstapel.

That still means the same thing, but now the location is emphasized. The verb liegt still stays in second position.

Yes.

The subject is der Nachziehstapel, which is singular, so the verb is also singular:

• der Nachziehstapel liegt = the draw pile lies / is located

If the subject were plural, the verb would change:

• Die Nachziehstapel liegen neben der Kartenauslage.
  The draw piles are next to the card display.

So:

• liegt = singular
• liegen = plural (or infinitive)

Neben is a preposition, so it normally comes before the noun phrase it governs:

• neben der Kartenauslage

That is the standard and expected structure in German.

What can move is the whole prepositional phrase:

• Der Nachziehstapel liegt neben der Kartenauslage.
• Neben der Kartenauslage liegt der Nachziehstapel.

But neben itself stays attached to its noun phrase.

A rough pronunciation guide is:

• Der = like dare, but shorter
• Nachziehstapel ≈ NAKH-tsee-stack-pel
• liegt = leekt
• neben = NAY-ben
• der = again like short dare
• Kartenauslage ≈ KAR-ten-ouse-lah-guh

A few notes:

• ch in Nach- is the throaty sound heard in Scottish loch
• ie in liegt is a long ee sound
• z in zieh is pronounced ts
• au in Auslage sounds like ow in cow

Not really in most everyday situations, because both Nachziehstapel and Kartenauslage are strongly tied to a game or card-related context.

Grammatically, the sentence is perfectly normal. But semantically, it sounds like something from:

• board-game instructions
• a rulebook
• a game setup explanation
• a card-game tutorial

So it is a very natural sentence in that context.