Learn Farsi: شهر تهران پایتخت کشور ایران است. - The city of Tehran is the capital of the country of Iran.

This is the Explanatory Ezafe (Ezafe-ye Bayâni). Instead of showing possession (like "Ali's book"), it links a general geographic noun like shahr (city) or keshvar (country) to its specific proper name. It works exactly like the English phrasing "the city of Tehran" or "the country of Iran."

Is it mandatory to use shahr-e and keshvar-e here?

No, you can easily just say Tehrân pâyetakht-e Irân ast (Tehran is the capital of Iran). Including the words for "city" and "country" using the Explanatory Ezafe just makes the sentence sound a bit more formal, descriptive, or encyclopedic, much like it does in English.

Is there an unwritten ezafe after pâyetakht?

Yes, you must pronounce it as pâyetakht-e keshvar-e Irân. Unlike the Explanatory Ezafes used for the geographic names in this sentence, this one is a standard possessive ezafe. It translates to "the capital of the country..." As always, this short vowel isn't written in the Persian script.

What is the literal breakdown of the word pâyetakht (capital)?

It is a compound of pâye (foot or base) and takht (throne). Historically, the "foot of the throne" was the city where the monarch lived and held court, which naturally became the Farsi word for "capital city."