"Delhi is a city."
Translation:दिल्ली एक शहर है।
No. Colloquially, people would say शहर.
नगर is a 'tatsam' word which means that it has been borrowed as is from Sanskrit. These words are seen as more formal in modern Hindi^* so you are more likely to see them in written form, in the names of organisations or in formal speech than in conversational Hindi.
^*- As opposed to words borrowed from Persian/Arabic and the other category of Sanskrit-derived words which are called 'Tadbhav'. These are words which have been part of Hindustani and its parent Prakrit languages for longer and have undergone some changes.