Sure it is. When you take a French course they have you learn the name "Etats-Unis" (or "Angleterre" or "Australie" or wherever you're from) as a French word along with all the others.
Yes. The name “Beijing” is for people writing in pinyin, which is not a transliteration designed for English speakers. English speakers should use Yale romanization or the traditional postal romanization for well known places.
I called it Pékin for years and as far as I know the pronunciation is closer to the Chinese name than Beijing
It’s not. Beijing is pretty accurate as most English speakers say it (except for the tones), although I often hear people do really weird things with the j. The b and j sounds in Mandarin are not the same as in English, but they are closer than p or k to my ear.