Learn Farsi: دکتر شتابان به بیمارستان رفت.

It is created by taking the present stem of the verb شتافتن (to hurry or hasten), which is شتاب, and adding the suffix -ان. This makes a present participle that shows the state or manner of the subject, translating to 'hurrying' or 'hastily'.

Is شتابان acting as an adjective or an adverb here?

In this specific sentence, it acts as an adverb describing how the doctor went to the hospital. These -ان words can often function as both adjectives and adverbs depending on where they are placed in the sentence.

Could I connect the doctor and the participle with an ezâfe, like دکترِ شتابان?

You could, but it changes the meaning slightly. Adding the ezâfe turns شتابان into a direct adjective modifying the doctor. دکترِ شتابان به بیمارستان رفت means 'The hurrying doctor went to the hospital,' whereas the original sentence means 'The doctor went to the hospital hastily.'

As a quick review, how is the word بیمارستان constructed?

It is made of the word بیمار (sick person or patient) plus the suffix -ستان (place of). So it literally means 'place of the sick.' You see this same suffix in words like قبرستان (cemetery) or country names like انگلستان (England).