Elle porte une casquette bleue.

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

In French, porter is commonly used for wearing clothes, accessories, glasses, hats, etc.

• Elle porte une casquette. = She is wearing a cap.
• Il porte un sac. = He is carrying a bag.

So the verb stays the same, and the object tells you which meaning is intended.

French usually requires an article before a singular countable noun.

So you normally say:

• une casquette = a cap
• la casquette = the cap

Unlike English, French does not usually allow a bare singular noun in this kind of sentence. That is why Elle porte casquette bleue would be incorrect.

Because casquette is a feminine noun.

In French, every noun has a grammatical gender, and the article must match it:

• un for masculine singular nouns
• une for feminine singular nouns

Since casquette is feminine, it takes une.

Because the adjective must agree with the noun it describes.

Here, casquette is feminine singular, so bleu changes to bleue.

Agreement pattern:

• masculine singular: bleu
• feminine singular: bleue
• masculine plural: bleus
• feminine plural: bleues

Examples:

• un chapeau bleu
• une casquette bleue
• des chapeaux bleus
• des casquettes bleues

In French, most adjectives come after the noun.

So:

• une casquette bleue = a blue cap

This is different from English, where adjectives usually come before the noun.

A small group of very common adjectives often come before the noun instead, such as those related to beauty, age, number, goodness, and size:

• un petit chien
• une belle maison

But color adjectives like bleu / bleue normally come after the noun.

French often uses the simple present tense where English may use either the simple present or the present progressive.

So:

• Elle porte une casquette bleue can mean She wears a blue cap or She is wearing a blue cap, depending on context.

French does have ways to emphasize an action in progress, such as être en train de, but that is not usually necessary here.

• Elle est en train de porter une casquette bleue is grammatically possible, but it sounds unnecessarily heavy in most normal contexts.

Yes. Elle usually means she, but it can also mean it when referring to a feminine noun.

For example:

• La voiture est neuve. Elle est bleue. = The car is new. It is blue.

In your sentence, if the subject is a person, elle means she.

Yes, but it means something slightly different.

• Elle porte une casquette bleue = She is wearing a blue cap.
• Elle a une casquette bleue = She has a blue cap / She owns a blue cap / She has one with her.

So porter focuses on wearing, while avoir focuses on possessing.

Casquette means a cap, usually with a visor. Chapeau is a more general word for hat.

So:

• une casquette = a cap
• un chapeau = a hat

If you specifically mean something like a baseball cap, casquette is the right word.

A simple approximate pronunciation is:

Ell port uhn kass-kett bluh

A few notes:

• Elle sounds like ell
• porte sounds roughly like port
• une has a French vowel that does not exist exactly in English; uhn is only an approximation
• casquette sounds like kass-kett
• bleue sounds like bluh with rounded lips

Also, there is normally no strong liaison between porte and une here in standard careful speech.

Because the noun is singular here: une casquette = one cap.

The adjective agrees with the noun in both gender and number:

• une casquette bleue = one feminine singular cap
• des casquettes bleues = multiple feminine plural caps

So bleue is singular because casquette is singular.

It depends on who they refers to.

For they referring to women or girls:

• Elles portent une casquette bleue. = They are wearing a blue cap.

For they referring to men or a mixed group:

• Ils portent une casquette bleue.

Notice that:

• elle becomes elles or ils
• porte becomes portent

If each person is wearing one blue cap, une casquette bleue can stay singular. If you mean blue caps in general, you might also see:

• Elles portent des casquettes bleues.

Yes, it is completely natural.

Elle porte une casquette bleue is a normal, everyday French sentence. It is simple and idiomatic, and it uses very standard grammar:

• subject pronoun: elle
• verb: porte
• article + noun: une casquette
• adjective agreeing with the noun: bleue

It is exactly the kind of sentence a learner should know how to build.