الباب خشب.

Because Arabic usually leaves out the verb to be in the present tense.

So الباب خشب literally looks like the door wood, but it means the door is wood / the door is made of wood.

This is very normal in both Standard Arabic and Egyptian Arabic.
If you want past or future, then Arabic does use a verb:

• الباب كان خشب = The door was wood / was made of wood
• الباب هيبقى خشب = The door will be wood / will be made of wood

In the present, though, no is is needed.

ال is the Arabic definite article, meaning the.

So:

• باب = a door / door
• الباب = the door

In Egyptian Arabic, ال is usually pronounced el-, so الباب is pronounced el-bāb.

Because خشب here is the predicate of the sentence, not another definite noun.

In a sentence like this, Arabic often has:

• a definite topic/subject first: الباب = the door
• then an indefinite predicate: خشب = wood

So الباب خشب means The door is wood / wooden.

If you put ال on both words, الباب الخشب, that would not work as this kind of sentence. It would not mean the door is wood.

It is basically a noun meaning wood.

Arabic often uses a noun like this after another noun to express material:

• الباب خشب = The door is wood
• more natural English: The door is made of wood or The door is wooden

So even though English often prefers an adjective like wooden, Arabic can simply use the noun wood.

Yes. خشبي means wooden.

So:

• الباب خشب = The door is wood / made of wood
• الباب خشبي = The door is wooden

Both are understandable, but they are slightly different in form:

• خشب = noun, wood
• خشبي = adjective, wooden

In Egyptian Arabic, using the material noun directly, as in الباب خشب, is very common and natural.

A common Egyptian pronunciation is:

el-bāb khashab

A few pronunciation notes:

• ال is pronounced el-
• باب has a long ā sound: bāb
• خ is the kh sound, like the ch in German Bach or Scottish loch
• خشب sounds like khashab

So the full sentence is roughly:

el-bāb khashab

Because Arabic is presenting the door as the topic first, then telling you something about it.

So the structure is:

• الباب = the thing we are talking about
• خشب = what we say about it

This is a very common pattern in Arabic nominal sentences.

If you change the order, the meaning or emphasis can change, and it may sound less neutral. For a basic beginner sentence, الباب خشب is the normal way to say it.

Not in this sentence.

Since خشب is a noun meaning wood, it stays the same here:

• الباب خشب = The door is wood
• الترابيزة خشب = The table is wood

But if you use the adjective خشبي instead, then it does agree:

• masculine: الباب خشبي
• feminine: الترابيزة خشبية

So:

• خشب as a material noun does not change here
• خشبي / خشبية as an adjective does change

It makes sense, but the most natural interpretation depends on context.

In Egyptian Arabic, الباب خشب is very natural and common for The door is made of wood.

In Standard Arabic, people may more often use something a bit fuller, such as:

• الباب من الخشب = The door is made of wood
• الباب خشبي = The door is wooden

So your sentence works very well for Egyptian-style everyday usage, even though the writing itself looks normal in standard Arabic script.

Because Arabic is usually written without short-vowel marks in normal everyday text.

So الباب خشب is written in a simple unvowelled form, but a speaker knows how to read it from context.

If you added full vowel marks, you might see something like:

• البابُ خشبٌ in a very formal Standard Arabic style with case endings

But in Egyptian Arabic, those case endings are not used in speech, and in normal writing people usually just write:

• الباب خشب

So what you see is completely normal.