Learn Polish: Czy ten nowy szpital miejski jest już otwarty? - Is this new city hospital already open?

This perfectly shows the difference between descriptive and classifying adjectives! Nowy (new) is a descriptive adjective giving an opinion or a temporary quality, so it goes before the noun. Miejski (city/municipal) is a classifying adjective that tells us the specific category or type of hospital, so it goes directly after the noun.

What if I changed the order to nowy miejski szpital? Would that be wrong?

Yes, it would sound unnatural to a Polish speaker. Because miejski classifies the hospital as a specific type (a city hospital), it belongs tightly attached after the noun. Putting it before the noun breaks that fixed category phrase.

Why is otwarty in the nominative case instead of the instrumental? I thought words following jest take the instrumental.

Great catch! The rule is that nouns following być (to be) take the instrumental case. However, when the word following być is an adjective describing the subject, like otwarty does here, it stays in the nominative case.

Is otwarty a regular adjective or a verb form?