Imperium Romanum or Imperium Romanorum. Imperium is a difficult to translate word that essentially means power and the right to command, so the Romans literally talked about the empire spatially as essentially being the area where the power of the Romans held sway. More rhetorically, they spoke of the orbis terrarum, or literally the circle of the earth, as the empire was frequently conceived of as universal.

Tradition holds that it was named after Romulus, of Romulus and Remus fame. They are the two legendary brothers who, according to tradition, were behind the founding of the city/kingdom in 753 BC. Romulus killed his brother and seized sole control. Likely, the name of Rome's first king was formed in retrospect as a later origin myth. We do know that Rome was a monarchy before the founding of the Republic in the late 6th century/ early 5th century B.C. The history of the monarchy is semi-mythical and Romulus is considered the first of seven kings.

Roman culture was initially heavily influenced by Etruscan culture, and the Etruscan word "rumon" might be a possible explanation for the name.

Sorry for ambiguity. How did the Romans call their country?

Before the Early Modern Period the concept of the nation-state didn't really exist. There were citizens, freedmen, and slaves, who lived in provinces, and were ruled by the Emperor and the Senate. Borders were far more fluid, and loyalty less well-defined than in the present.

I am not sure what you're asking?

"Roma, Romae" is the Latin name.
"Urbe, Urbis" would most often refer to the city of rome, because Rome was the city.