Unter dem Tisch ist ein Loch im Teppich.

Because unter is a two-way preposition in German. That means it can take either:

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

In Unter dem Tisch ist ein Loch im Teppich, the table is just the location, so German uses the dative:

• unter dem Tisch = under the table

Compare:

• Das Buch liegt unter dem Tisch. = The book is under the table.
• Ich lege das Buch unter den Tisch. = I put the book under the table.

So dem is dative because there is no movement involved.

German word order is flexible, and the sentence often begins with the part the speaker wants to emphasize or set as the scene.

Here, Unter dem Tisch comes first to establish the location:

• Unter dem Tisch = under the table

Then the verb comes in the second position, as German main clauses usually require:

• Unter dem Tisch ist ...

So this is normal German word order. It sounds natural because it first tells you where to look, then tells you what is there.

Other orders are possible, such as:

• Ein Loch im Teppich ist unter dem Tisch.

But that version sounds less natural in many contexts.

Because ein Loch is the subject of the sentence, and subjects are in the nominative case.

Loch is a neuter noun: das Loch.

So the nominative indefinite article is:

• ein Loch

Not:

• einen Loch — this is wrong
• einen is used for masculine accusative, for example: Ich sehe einen Tisch.

Even though ein Loch comes after the verb, it is still the subject:

• Unter dem Tisch ist ein Loch im Teppich.

German does not rely only on word order to show the subject; case is very important.

im is a contraction of:

• in dem → im

So:

• im Teppich = in dem Teppich

This contraction is extremely common and usually preferred when it is possible.

Examples:

• im Haus = in the house
• im Garten = in the garden
• im Teppich = in the carpet

Because a hole is understood as being in the material of the carpet, not just on its surface.

So:

• ein Loch im Teppich = a hole in the carpet

If you said auf dem Teppich, that would mean on the carpet, as something lying on top of it.

Compare:

• Ein Fleck ist auf dem Teppich. = A stain is on the carpet.
• Ein Loch ist im Teppich. = A hole is in the carpet.

English also usually says a hole in the carpet, so this matches closely.

Because unter and unten are different kinds of words.

• unter is a preposition and must be followed by a noun phrase:
  • unter dem Tisch = under the table
• unten is an adverb meaning down below or at the bottom:
  • Das Loch ist unten. = The hole is down below.

So in this sentence you need unter, because it introduces dem Tisch.

Because in German, all nouns are capitalized.

So:

• der Tisch
• das Loch
• der Teppich

This is a standard rule of German spelling, not something special about this sentence.

The dictionary forms are:

• der Tisch = the table
• das Loch = the hole
• der Teppich = the carpet

That matters because gender affects the articles and case endings:

• unter dem Tisch
  
  • Tisch is masculine, dative singular → dem
• ein Loch
  
  • Loch is neuter, nominative singular → ein
• im Teppich = in dem Teppich
  
  • Teppich is masculine, dative singular → dem

Unfortunately, grammatical gender often has to be learned with each noun.

Yes, that is possible, but it means something slightly different in structure and emphasis.

• Unter dem Tisch ist ein Loch im Teppich.
  This presents the location first: Under the table, there is a hole in the carpet.

• Es gibt ein Loch im Teppich unter dem Tisch.
  This means There is a hole in the carpet under the table, but es gibt is a more general there is/there are construction.

In everyday German, the version with ist often sounds more direct and natural when describing where something is.

In meaning, unter dem Tisch tells you the location of the whole situation: the hole is located under the table.

So semantically, it describes where the hole is:

• the hole is under the table

Grammatically, it functions as an adverbial phrase at the start of the sentence.

The sentence can be understood as:

• [Under the table] [is] [a hole in the carpet].

So it is not part of the noun phrase ein Loch im Teppich; it is a separate location phrase.

Because German often does not need an equivalent of English there is.

English says:

• There is a hole in the carpet.

German often simply says:

• Ein Loch ist im Teppich.
• Im Teppich ist ein Loch.

So in this sentence:

• Unter dem Tisch ist ein Loch im Teppich.

German just uses ist without adding a dummy word like English there. This is very common.

Yes. German allows several natural variations, depending on emphasis.

For example:

• Im Teppich ist unter dem Tisch ein Loch.
• Ein Loch ist unter dem Tisch im Teppich.
• Unter dem Tisch ist im Teppich ein Loch.

But not all versions sound equally natural in every context.

The original:

• Unter dem Tisch ist ein Loch im Teppich.

is very natural because it first gives the place under the table, then introduces the new thing you notice: a hole in the carpet.