مقود الدراجة جديد.

Because this is a nominal sentence in Arabic. In the present tense, Arabic usually does not use a separate word for is / am / are.

So:

• مقود الدراجة جديد = The bicycle’s handlebar is new

In past or future contexts, Arabic can use verbs, but in a simple present-tense statement like this, the linking verb is normally omitted.

This is a nominal sentence:

• مقود الدراجة = the subject
• جديد = the predicate

In traditional grammar terms:

• مقود الدراجة = مبتدأ
• جديد = خبر

So the pattern is basically:

• [subject] + [description]

which gives: The bicycle’s handlebar [is] new.

Because مقود الدراجة is an iḍāfa construction, often called a genitive construction or noun-noun possession structure.

In an iḍāfa:

• the first noun does not take الـ
• the second noun can be definite or indefinite
• the whole phrase gets its definiteness from the second noun

So here:

• مقود = handlebar
• الدراجة = the bicycle

Together:

• مقود الدراجة = the bicycle’s handlebar / the handlebar of the bicycle

Even though مقود has no الـ, the whole phrase is definite because الدراجة is definite.

It is definite.

That is one of the key points of the iḍāfa construction:

• مقود دراجة = a bicycle’s handlebar / the handlebar of a bicycle → indefinite
• مقود الدراجة = the bicycle’s handlebar → definite

So in your sentence, the subject is definite.

Because جديد here is not an adjective inside the noun phrase. It is the predicate of the sentence.

Compare these:

• مقود الدراجة الجديد = the bicycle’s new handlebar
  Here الجديد is an adjective describing the noun phrase.

• مقود الدراجة جديد = The bicycle’s handlebar is new
  Here جديد is the predicate: it tells us something about the subject.

Predicate adjectives in sentences like this are normally indefinite in Arabic, so جديد does not take الـ.

Because it agrees with مقود, which is:

• masculine
• singular

So the predicate is also masculine singular:

• مقود ... جديد

If the noun were feminine, you would expect جديدة instead.
If it were plural, the form could change depending on the type of plural.

It agrees with مقود, not with الدراجة.

That is because مقود is the head noun of the subject phrase:

• مقود الدراجة = the bicycle’s handlebar

The thing being described as new is the handlebar, not the bicycle.

So:

• مقود = masculine singular
• therefore جديد = masculine singular

In fully vocalized formal Arabic, it would be:

• مِقْوَدُ الدَّرَّاجَةِ جَدِيدٌ

Case-wise:

• مِقْوَدُ → nominative, because it is the subject of the nominal sentence
• الدَّرَّاجَةِ → genitive, because it is the second term of the iḍāfa
• جَدِيدٌ → nominative, because it is the predicate

In normal writing, these short vowel endings are usually omitted.

Because the ل of الـ assimilates before certain letters called sun letters.

The word دراجة starts with د, and د is a sun letter. So:

• written: الدراجة
• pronounced: ad-darrāja(h)

This is a pronunciation change, not a spelling change.

Yes, but that means something different.

• مقود الدراجة جديد = The bicycle’s handlebar is new
• مقود الدراجة الجديدة = the handlebar of the new bicycle

In الدراجة الجديدة, الجديدة is an adjective describing الدراجة.

So adding الجديدة changes the noun phrase itself, not the predicate of the sentence.

Yes, but the meaning becomes less specific unless the context already makes the bicycle clear.

• المقود جديد = The handlebar is new
• مقود الدراجة جديد = The bicycle’s handlebar is new

So your original sentence is more explicit because it identifies which handlebar.

Yes. The iḍāfa construction is one of the most common and natural ways to express possession or a close relationship between two nouns.

So instead of using a separate word like of or ’s, Arabic often simply places two nouns together:

• مقود الدراجة = the bicycle’s handlebar
• literally: handlebar the-bicycle

This is a very common pattern, and learning it early is extremely useful.