Der Hund ist mein Liebling.

Because der Hund is the subject of the sentence, and subjects are in the nominative case.

• der Hund = nominative
• den Hund = accusative
• dem Hund = dative

In Der Hund ist mein Liebling, the dog is the thing being talked about as being something, so nominative is correct.

In a normal German main clause, the conjugated verb usually comes in second position. This is one of the most important German word-order rules.

Here the structure is:

• Der Hund = first element
• ist = second element
• mein Liebling = the rest

So the sentence follows the standard German pattern: subject + verb + complement.

Because mein is a possessive determiner, and its ending depends on the gender, number, and case of the noun that follows.

Here:

• Liebling is masculine
• it is singular
• it is in the nominative after sein (ist)

For a masculine singular noun in the nominative, mein takes no ending:

• mein Liebling
• compare: meine Katze (feminine)
• compare: mein Kind (neuter)

So mein Liebling is the correct form.

Because mein already does the job of an article-like word. In German, possessives such as mein, dein, sein, ihr, etc. usually replace the article.

So you normally say:

• mein Liebling = my favorite / my darling
• der Liebling = the favorite

but not usually:

• der mein Liebling

No. Liebling can be used in more than one way.

Common uses include:

• favorite: the person, animal, or thing you like best
• darling / sweetheart: as a term of affection

In this sentence, mein Liebling most naturally means something like my favorite or the one I love most, especially if you are talking about a pet.

So it is not necessarily romantic.

Not necessarily. Liebling is grammatically masculine as a noun, but that does not automatically mean the real-life animal is male.

German grammatical gender and biological sex are not always the same thing.

So even if the pet were female, the noun Liebling would still usually stay masculine:

• Die Hündin ist mein Liebling.

That is grammatically possible because Hündin is feminine, but Liebling is still a masculine noun.

Because in German, all nouns are capitalized.

So:

• Hund is a noun
• Liebling is a noun

That is why both begin with capital letters, even in the middle of a sentence.

Yes, you can. That sentence is also grammatical, but the emphasis changes.

• Der Hund ist mein Liebling.
  This starts with the dog and then says what he is.

• Mein Liebling ist der Hund.
  This puts more focus on my favorite, as if you are identifying which one it is.

Both are correct; they just highlight different parts of the sentence.

It is in the nominative too.

After forms of sein such as ist, German uses a predicate noun or predicate nominative, not an accusative object.

So in:

• Der Hund ist mein Liebling

both der Hund and mein Liebling are nominative in function.

This is similar to English in older or more formal grammar, where people talk about It is I rather than It is me. German strongly keeps that nominative pattern after sein.