If I remember correctly, the name "Bizantine Empire" was first use by a 19th century historian.
how did they call themselfs during their long history?
How did the other Medieval powers (both European and Muslim) call it?
To what they called their Empire/Nation, I do not know. But they do call themselves rhomaioi, meaning "Roman". So they still considered themselves Roman, as they were the eastern half of the Roman Empire.
Source (Pg 87 has some more info on this topic, if only a little)