Learn Hardcore Japanese: syokudou no doa ha aoi desu.

The の here is a genitive (possessive) linker. It connects 食堂 (the dining hall) to ドア (door) to mean “the dining-hall’s door” or simply “the dining-hall door.”

Why is は used after ドア instead of が?

は is the topic marker. By saying 食堂のドアは…, you’re making “the dining-hall door” the topic of your sentence (“As for the dining-hall door…”). If you used が (食堂のドアが青いです), you’d be simply stating “the dining-hall door exists/is blue,” focusing on new information rather than setting a topic.

Why is the adjective 青い followed by です? Isn’t 青い enough to make a sentence?

In casual (dictionary) style, you can end with 青い by itself:
食堂のドアは青い。
But in polite style you add the copula です after an i-adjective, so 青いです makes the sentence more formal.

Can I drop です after 青い if I want to sound casual?

Yes. In conversational or plain form you omit です:
食堂のドアは青い。
Use 青いです when speaking politely to teachers, strangers, or in formal settings.

What’s the difference between 青 and 青い?

青い is an i-adjective meaning “blue.”
青 by itself is a noun meaning “the color blue.”
If you said 食堂のドアは青です, you’d be treating 青 as a noun (“As for the door, it is the color blue.”). Using 青いです is more natural when describing something’s color.

Why do we need の between 食堂 and ドア? Can’t we just say 食堂ドア?

Japanese nouns don’t combine like English compounds; you need の to link them. While you might see abbreviations in signs, grammatically you must say 食堂のドア to show that the door belongs to the dining hall.

Could I instead say 食堂はドアが青いです? How would that change the focus?

Yes. 食堂はドアが青いです means “As for the dining hall, its door is blue.”

In 食堂のドアは青いです, the topic is the door itself.
In 食堂はドアが青いです, the topic is the dining hall, and you’re commenting on one of its features (the door).

What is the basic word order in this sentence?

Japanese follows:
[Topic] は → [Predicate] → です (optional polite copula)
Here:

食堂のドア (topic phrase)
は (topic marker)
青い (predicate = i-adjective)
です (polite ending)