Ik draag de hoed.

Because Dutch verbs are conjugated to match the subject.

The infinitive is dragen = to wear / to carry.
With ik (I), the verb usually loses the -en ending:

• ik draag = I wear / I carry
• jij draagt = you wear / carry
• hij draagt = he wears / carries

So ik draag de hoed is the correct present-tense form.

Dragen can mean several related things in Dutch, including:

• to wear clothes, glasses, jewelry, a hat, etc.
• to carry
• sometimes to bear or support

In Ik draag de hoed, the most natural meaning is I am wearing the hat or I wear the hat.

So unlike English, Dutch uses the same verb for both wearing something on your body and carrying something in some contexts.

Because hoed is a de-word in Dutch.

Dutch nouns take either:

• de
• het

You usually have to learn this with each noun.
So:

• de hoed = the hat

There is not always a simple logic you can see from the meaning alone, so it is best to memorize nouns together with their article.

Normally, no. In Dutch, as in English, singular countable nouns usually need an article or another determiner.

So you say:

• Ik draag de hoed = I wear the hat
• Ik draag een hoed = I wear a hat
• Ik draag mijn hoed = I wear my hat

But Ik draag hoed sounds incomplete and ungrammatical in standard Dutch.

The difference is the same basic idea as the hat versus a hat in English.

• de hoed = the hat
  A specific hat, one that speaker and listener can identify.
• een hoed = a hat
  Any hat, or a hat not yet specifically identified.

So:

• Ik draag de hoed = I’m wearing the hat
• Ik draag een hoed = I’m wearing a hat

Yes. Dutch simple present often covers both meanings.

So Ik draag de hoed can mean:

• I wear the hat
• I am wearing the hat

Context tells you which is intended.

Dutch does have ways to emphasize an ongoing action, but in everyday speech the simple present is very often enough.

Because Dutch main clauses usually follow the verb-second pattern.

In a normal statement:

• first comes the subject or another element
• the finite verb comes early in the sentence, typically in second position

So here:

• Ik = subject
• draag = finite verb
• de hoed = object

That gives:

• Ik draag de hoed

This is very similar to basic English word order in a simple statement.

In a yes/no question, Dutch usually puts the verb first:

• Draag ik de hoed? = Am I wearing the hat? / Do I wear the hat?

So the order changes from:

• Ik draag de hoed.

to:

• Draag ik de hoed?

Usually with niet:

• Ik draag de hoed niet. = I am not wearing the hat.

In this sentence, niet normally goes near the end, after the object.

So:

• Ik draag de hoed. = I wear the hat.
• Ik draag de hoed niet. = I do not wear the hat.

Roughly, draag sounds something like drahkh.

A few notes:

• aa is a long vowel, somewhat like the a in father for many speakers.
• The final g is not like the English g in go. In standard Dutch it is usually a throaty sound, like the ch in Scottish loch or German Bach.
• The r varies by region.

So do not pronounce it like English drag. The Dutch word sounds different, especially at the end.

The dictionary form is dragen.

Useful present-tense forms are:

• ik draag
• jij draagt
• u draagt
• hij/zij/het draagt
• wij dragen
• jullie dragen
• zij dragen

So when you learn Ik draag de hoed, it is helpful to connect draag with the infinitive dragen.

Usually hoed means a hat, especially the kind with a brim.

Examples:

• een hoed = a hat
• een pet = a cap

So if you mean something like a baseball cap, Dutch often uses pet, not hoed. But for the sentence Ik draag de hoed, hoed is perfectly normal for hat.