De asielhond draagt een muilkorf.

Because the main noun is hond, and hond is a de-word.

In Dutch compound nouns, the last part usually determines the grammatical gender and article:

• het asiel = the shelter
• de hond = the dog
• de asielhond = the shelter dog

So even though asiel takes het, the full compound takes de because hond is the head word.

Literally, it is made up of:

• asiel = shelter
• hond = dog

So asielhond means shelter dog or dog from an animal shelter.

Dutch often forms one compound word where English might use two words or a phrase.

Because Dutch writes compound nouns as one word much more often than English does.

So:

• asiel + hond → asielhond
• muil + korf → muilkorf

This is very normal in Dutch. English often separates these words, but Dutch usually combines them.

Because the subject is singular: de asielhond = the shelter dog.

The infinitive is dragen = to wear / to carry.
For hij/zij/het or any singular noun in the present tense, Dutch usually adds -t:

• ik draag = I wear/carry
• jij draagt = you wear/carry
• hij/zij/het draagt = he/she/it wears/carries

So:

• De asielhond draagt een muilkorf.

Yes. Dragen can mean both to carry and to wear, depending on context.

In this sentence, the dog is not carrying the muzzle in its paws or mouth. It is wearing it, so dragen is best understood as to wear.

Dutch commonly uses dragen for wearing:

• een jas dragen = to wear a coat
• een bril dragen = to wear glasses
• een muilkorf dragen = to wear a muzzle

Because een means a/an, while de means the.

So:

• een muilkorf = a muzzle
• de muilkorf = the muzzle

The sentence uses the indefinite article because it is introducing the item as a muzzle, not referring to a specific one already known in the conversation.

Also, muilkorf is a de-word, so its definite form is de muilkorf.

Muilkorf literally breaks down as:

• muil = snout / muzzle / animal mouth
• korf = basket

Together, muilkorf means muzzle.

It is the normal Dutch word for the device put over a dog’s mouth to prevent biting.

Yes. This is the standard subject–verb–object order used in a simple main clause:

• De asielhond = subject
• draagt = finite verb
• een muilkorf = object

So the sentence structure is very straightforward: The shelter dog wears a muzzle.

Yes, absolutely.

That version means something like: The dog from the shelter is wearing a muzzle.

The difference is mainly style:

• de asielhond = more compact, like the shelter dog
• de hond uit het asiel = more explicit, like the dog from the shelter

Both are natural, but asielhond is a neat compound noun.

A rough English-friendly guide:

• asielhond ≈ ah-SEEL-hont
• muilkorf ≈ MOWL-korf

A few notes:

• asiel has stress on the second syllable: a-SIEL
• the ui in muil is a difficult Dutch vowel that has no exact English equivalent
• final d in hond is often pronounced more like t at the end of the word

The exact pronunciation depends on accent, but those rough guides can help you get started.

The plurals are:

• de asielhond → de asielhonden
• de muilkorf → de muilkorven

So you could say:

• De asielhonden dragen muilkorven.
  = The shelter dogs wear muzzles.

Notice:

• singular indefinite: een muilkorf
• plural indefinite: just muilkorven with no article if you mean muzzles in general

Because it says de asielhond, it refers to a specific shelter dog in context, or to the shelter dog as an identifiable one.

If you wanted to mean a shelter dog, you would say:

• Een asielhond draagt een muilkorf.

That could mean A shelter dog wears a muzzle or A shelter dog is wearing a muzzle, depending on context.